ADE Little String Theory

on a Riemann Surface

(and Triality)

Mina Aganagic1,2, Nathan Haouzi1

1Center for Theoretical Physics and 2Department of Mathematics

University of California, Berkeley, USA

Abstract

We initiate the study of (2, 0) little string theory of ADE type using its

definition in terms of IIB string compactified on an ADE singularity. As one

application, we derive a 5d ADE quiver gauge theory that describes the little

string compactified on a sphere with three full punctures, at low energies.

As a second application, we show the partition function of this theory equals

the 3-point conformal block of ADE Toda CFT, q-deformed. To establish

this, we generalize the An triality of [1] to all ADE Lie algebras; IIB string

perspective is crucial for this as well.

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1 Introduction

String theory has led to remarkable insights into quantum field theories in

various dimensions and many areas of mathematics. Recently, a plethora

of new conjectures about properties of gauge theories were obtained from

studying the six-dimensional (2, 0) CFT, whose existence is implied by string

and M-theory. The 6d theory seems poised to play an important role in

mathematics as well, see e.g. [2–7]. Yet, the lack of a good description of the

(2, 0) theory limits what predictions we are able to obtain.

The (2, 0) CFT is labeled by an ADE Lie algebra g and defined as the

limit of IIB string theory on an ADE surface X where we send the string

coupling to zero to decouple the bulk modes, and then ms to infinity to get a

CFT. If we keep the string scale finite instead, the result is a six-dimensional

string theory, the (2, 0) little string theory [8–10]. The description of the

little string theory in terms of IIB string on an ADE singularity in the limit

of zero string coupling gives us an essentially complete control of the theory,

away from the singularity at the origin of its moduli space [8].

In this paper, we initiate the study of (2, 0) little string theory, in con-

nection to gauge theories and mathematics. This perspective is fruitful for

the 6d CFT as well: we simply take the string scale to infinity in the very

end. For illustration, we will give several applications of this approach.

We place the (2, 0) little string on a Riemann surface C which is a cylinder,

with co-dimension two defects at points of C. We identify the defects, using

known facts about IIB string on ADE singularities, as D5 branes wrapping

the (non-compact) 2-cycles of X. We show that the codimension two defects

preserving superconformal invariance can be classified using the relation [11]

between the geometry of X and the representation theory of g. We argue

that, except at the very origin of the moduli space, the (2, 0) little string

theory with defects has a low energy description as a 5d N = 1 ADE quiver

1

theory compactified on a (T-dual) circle. The theory is five-dimensional due

to string winding modes around the compact direction in C. It can be de-

termined by perturbative IIB string analysis [12] for any collection of defects

introduced by D5 branes. As an example, we generalize the gauge theory

description of the so called 5d TN theory, which was recently obtained in the

A-series case in [1, 13, 14], to all ADE groups. The gauge theory descrip-

tion of the low energy physics lets us obtain the supersymmetric partition

function of the little string on C × R4, with R4 regulated by Ω-background,

using techniques of [15–20]. This should be contrasted with the 6d CFT on

C, which generically has no Lagrangian description, so there is no direct way

to obtain the partition function. We also give a simple derivation of the

Seiberg-Witten curve of the theory by compactifying on an additional cir-

cle, and using T-duality to relate D5 branes to monopoles of an ADE gauge

theory on R× T 2, studied in [19, 20].

The second application is to the conjectural relation between the (2, 0)

6d CFT and 2d conformal field theory on C [5, 21–23]. We will show that the

partition function of the ADE little string on C × R4 equals the q-deformed

conformal block of ADE Toda CFT on C. The q-deformed vertex operator

insertions are in one to one correspondence with the defect D5 branes. The

q-deformed conformal blocks have a Wq,t(g) algebra symmetry developed by

Frenkel and Reshetikhin [24]. They are written as integrals over positions of

screening currents, with integrand which is a correlator in a free theory. We

show that, computing the integrals by residues, one recovers the partition

function of the little string. The numbers of screening charge integrals are

related to the values of the Coulomb moduli. In the limit in which we take

the string scale to infinity, and little string becomes the (2, 0) CFT on C, the

q-deformation disappears and one recovers ordinary Toda conformal block,

with W(g) algebra symmetry. The relationship between the (2, 0) CFT of

2

A1 type and Liouville theory (the A1 Toda CFT) was conjectured by Alday,

Gaiotto and Tachikawa, and the correspondence was proven in [25, 26]. The

An version of the correspondence was established in [1, 27]. The strategy

of the proof employed here is the same as in [1], although the string theory

realization of (2, 0) theory used there is different, related by T-duality.

We show, in a related application, that there is a triality of precise re-

lations between three different classes of theories of ADE type: the quiver

gauge theories in three and five dimensions and the q-deformed Toda CFT.

In establishing the correspondence between Toda CFT and the (2, 0) theory

we described, the central role is played by strings, obtained by wrapping D3

branes on compact 2-cycles in X, and at points on C. The D3 branes are fi-

nite tension excitations – they are vortices on the Higgs branch of little string

theory on C. The theory on D3 branes, derived by a perturbative IIB compu-

tation, is a 3d ADE quiver theory compactified on an S1, which in presence

of defect D5 branes has N = 2 supersymmetry. The 3d theory turns out to

have a manifest relation to Toda CFT: its partition function, expressed as

an integral over Coulomb moduli, is identical to the q-Toda CFT conformal

block – D3 branes are the screening charges. Interpreting the partition func-

tion instead in terms of the Higgs branch of the 3d gauge theory, it equals the

5d partition function, at integer values of Coulomb moduli. Physics of this

is the gauge/vortex duality which originates from two different, yet equiva-

lent ways to describe vortices: from the perspective of the theory on the D3

branes or from the perspective of the bulk theory with fluxes. In the latter

description, the vortex flux is responsible for shifting the Coulomb moduli

by integer values [1, 28] in Ω-background. This is the little string version of

large N duality of [29–32].

The paper is organized as follows. In section 2, we study the ADE (2, 0)

little string theory on C with co-dimension 2 defects, and solve it. In section

3

3, we study vortices in this theory, in terms of D3 branes, and show how to

solve this theory as well. In section 4, we review the free field formulation

of ADE Toda CFT, and their q-deformation. We show that the q-deformed

conformal block is the D3 brane partition function. In section 5, we review

gauge/vortex duality, and show that the partition function of the ADE (2, 0)

little string on C is the q-deformed ADE conformal block on C. In section 6,

we give examples corresponding to the 3-punctured sphere. We review [24]

in appendix A.

2 (2, 0) Little String Theory on C

ADE little string theory with (2, 0) supersymmetry is a six dimensional string

theory. It is defined by starting with IIB string theory on an ADE surface

X, in the limit where we take the string coupling to zero to decouple the

bulk modes [8–10].1 The surface X is a hyperkahler manifold, obtained by

resolving a C2/Γ singularity where Γ a discrete subgroup of SU(2), related

to g by McKay correspondence. The limit leaves a six-dimensional theory

supported near the singularity. The little string theory is not a local quantum

field theory. It contains strings whose tension is m2s and has a T-duality

symmetry that exchanges (2, 0) and (1, 1) little string, compactified on a

circle. The latter is obtained from IIA string on X, in the gs to zero limit.

At energies far below the string scale ms, the (2, 0) little string reduces to

(2, 0) 6d conformal field theory, studied in [3–7] and elsewhere. The little

string breaks the conformal invariance of the (2, 0) CFT, but it does so in a

canonical way.

The (2, 0) theory contains a non-abelian self-dual tensor field based on

the Lie algebra g, but no gravity. The moduli space of the little string theory

1See [33, 34] for review of little string theory. Reviews of many facts about Lie algebras,ADE singularities, quiver gauge theories and monopoles are in [19].

4

is

M = (R4 × S1)n/W, (2.1)

where n is the rank, and W the Weyl group, of g. The scalar fields parame-

terizingM come from the moduli of the metric on X; they are encoded in the

periods of a triplet of self-dual two-forms ωI,J,K and the NS and RR B-fields,

along the 2-cycles Sa generating H2(X,Z). Their natural normalizations are∫Sa

m4s ωI,J,K/gs,

∫Sa

m2s BNS/gs,

∫Sa

m2s BRR. (2.2)

A power of gs accompanies NS sector fields but not the RR sector ones, since

this is how they enter the low energy action of IIB string. The canonical mass

dimension of scalars in a two-form theory is two. In taking gs to zero we tune

the moduli of IIB so that the above combinations are kept fixed. The compact

directions inM come from periods of the RR B-field and have radius m2s. In

the low energy limit, when we send ms to infinity, the moduli space becomes

simply (R5)n/W , since periodicity of the scalars coming from BRR becomes

infinite. The periodicity of scalars coming from BNS would have been m2s/gs;

it is lost at the outset since gs is taken to zero.

The perturbative string theory description is good away from the singu-

larity at the origin of the moduli space M. We are taking gs to zero, in

addition, so from IIB string perspective the theory is under excellent control.

To obtain results pertaining to the (2, 0) CFT, we will want to take the ms

to infinity limit, but only at the very end.2

2.1 Little String on C with Codimension Two Defects

We compactify the (2, 0) little sting theory on a Riemann surface C. Here,

we will only consider the simplest possibility, where C is a cylinder with flat

2The singularity of the world sheet CFT that emerges at the origin ofM was describedin [8]. We will not need to worry about the breakdown of perturbation theory, as we willstay away from this point.

5

metric, so that X × C is a solution of IIB string theory. We would like to

introduce codimension two defects in the little string theory, which are at

points on C and fill the remaining 4 directions. In the little string limit, IIB

string has an essentially unique candidate for such an object: these are D5

branes that wrap non-compact 2-cycles in X, which are points on C, and

fill the rest of the space time.3 The other candidates either have infinite

tension as we take gs to zero, or decouple from the degrees of freedom of the

little string. The choice of 2-cycles in X for D5 branes to wrap are governed

by symmetries we would like to preserve. We will choose to preserve half

the supersymmetry and conformal invariance of the low energy theory. To

translate this into a geometric condition on 2-cycles, we need to first recall

some elements of the geometry of ALE spaces [11, 37]. Codimension two

defects of the (2, 0) CFT have been studied by different means in [3, 38–41].

2.1.1 Elements of Holomogy

The second homology group H2(X,Z) of X is a lattice whose generators are

n 2-spheres Sa, supported at the singularity of X. The 2-spheres intersect

according to the Dynkin diagram of g:

#(Sa ∩ Sb) = −Cab. (2.3)

The Dynkin matrix Cab is given in terms of the adjacency matrix Iab of the

Dynkin diagram as follows

Cab = 2δab − Iab.

As a lattice, H2(X,Z) is the same as the root lattice Λ of g: the homology

classes of cycles Sa correspond to simple positive roots ea of the Lie algebra,

the intersection pairing is the inner product on Λ, up to an overall sign.

3String-like defects in 6d SCFTs were given an analogous description in [35], replacingD5 with D3 branes. This leads to degenerate vertex operators of Wq,t(g) algebra [36].

6

The relative homology group H2(X, ∂X,Z) of X corresponds [37] to the

weight lattice Λ∗ of g. Here, one allows 2-cycles with boundary at infinity ∂X.

A two cycle is trivial in H2(X, ∂X,Z) if it is a boundary of a three-cycle in X

up to addition of a 2-cycle at infinity ∂X. The group contains the H2(X,Z)

as a sub-lattice, by restricting to compact 2-cycles. Correspondingly, the root

lattice is a sub-lattice of the weight lattice Λ ⊂ Λ∗. The group is spanned by

classes of non-compact 2-cycles S∗a, which are, up to sign, the fundamental

weights wa of g. The fundamental weights are defined by (ea, wb) = δab, so

that

#(Sa ∩ S∗b ) = δab. (2.4)

To construct a cycle in the homology class of S∗b , one starts by zooming in on

the neighborhood of the vanishing 2-cycle Sb, which is locally T ∗S2. Next,

one picks a point on Sb, away from intersections with other minimal 2-cycles

and takes S∗b to be the fiber of the cotangent bundle to Sb above that point.

Per construction, this satisfies (2.4), and extends to the boundary of X. We

distinguish the cycles in X from their homology classes, denoted by [..]. For

example, the homology class of the a-th vanishing 2-cycle Sa is the simple

root [Sa] = ea, and a class of the dual non-compact cycle S∗a is minus the

fundamental weight [S∗a] = −wa.

Figure 1: The vanishing cycles of An singularity Sa (in black) and the dualnon-compact cycles S∗a (in blue). For any ADE singularity, S∗a is constructedas the fiber of the cotangent bundle T ∗Sa over a generic point on Sa.

7

2.1.2 Homology Classes of Defects

.

To specify a defect D5 brane charge, we pick a collection of non-compact

two-cycles S∗ whose homology class in H2(X, ∂X,Z) = Λ∗ is

[S∗] = −n∑a=1

mawa ∈ Λ∗ (2.5)

with positive integer coefficients ma. A-priori, m’s can be arbitrary, but we

would like to preserve conformal invariance in 4d, which imposes constraints.

A necessary condition for conformal invariance is that the net D5 brane flux

vanishes: a non-zero flux of HRR would lead to varying periods of BRR (and

running of the gauge coupling constant on the D5 branes). To satisfy the

condition, we add D5 branes that wrap a compact homology class [S] ∈H2(X,Z) = Λ:

[S] =n∑a=1

da ea ∈ Λ (2.6)

where d’s are also non-negative integers, such that

[S + S∗] = 0. (2.7)

In adding (2.6) and (2.5), we used that the root lattice embeds into the

weight lattice and the compact homology group into the relative homology

group by considering cycles with trivial boundary. The vanishing of S + S∗

in homology implies its intersection with any 2-cycle Sa vanishes. With help

of (2.3) and (2.4), we can write this as

n∑b=1

Cab db = ma. (2.8)

To preserve supersymmetry, it is not enough to choose the class of S∗, we

must choose the actual cycles in it. D5 branes wrapping different components

of S∗ preserve the same supersymmetry if their central charges are aligned.

8

These, in turn, are determined by the periods of the triplet of self-dual two

forms ~ω = (ωI , ωJ , ωk) on the non-compact cycles S∗a. Supersymmetry is

preserved if they determine a collection of vectors,∫S∗b~ω, which point in

the same direction for all b, corresponding to all the central charges being

aligned. Then, all the non-compact D5 branes preserve the same half of

supersymmetries of the (2, 0) theory. Up to a rotation under which ~ω is a

vector, we can choose ∫S∗a

ωI > 0,

∫S∗a

ωJ,K = 0.

Next, we pick a metric on X by picking periods of ωI,J,K through the compact

cycles Sa. The choice we make will affect the supersymmetry that D5 branes

wrapping compact 2-cycles preserve. It does not affect the non-compact D5

branes, which extend to infinity in X, as it only affects the data of X near

the singularity. We will begin by setting∫Sa

ωJ,K = 0,

∫Sa

BNS = 0, (2.9)

for all a’s and letting

τa =

∫Sa

(m2s ωI/gs + i BRR) (2.10)

be arbitrary complex numbers with Re(τa) > 0. Recall that X has a sphere’s

worth of choices of complex structure. In the complex structure in which

ωI is a (1, 1) form, and having chosen (2.9), (2.10), both [S∗b ] and [Sa] have

holomorphic 2-cycles representatives, and the D5 branes wrapping both the

compact and the non-compact 2-cycles preserve the same supersymmetry.

The fact that all the D5 branes preserve the same supersymmetry is impor-

tant, as it leads to an ADE quiver gauge theory description at low energies,

with the quiver diagram based on the Dynkin diagram of g.

9

2.1.3 Gauge Theory Description of The System

We will now determine the low energy description of the compactified (2, 0)

little string with defects. For generic τ ’s, at energies below the string scale,

the entire system can be described in terms of a 5d N = 1 gauge theory

which originates from the D5 branes.

At long distances, if τ ’s are not zero, the bulk theory is a theory of abelian

self-dual 2-forms. The 2-forms are non-dynamical from the perspective of

the compactified theory, since they propagate in all six dimensions. At the

same time, τ ’s determine the inverse gauge couplings of the D5 brane gauge

theory. As long as they are non-zero, the theory on the D5 branes has a

gauge theory description at low energies.4 Thus, for non-zero τ and below

the string scale, the dynamics of the (2, 0) little string theory on C with

defects can be described by the gauge theory on the D5 branes.

String theory allows us to determine the gauge theory on the D5 branes

on X. The gauge theory on the D5 branes wrapping S was worked out in

[12]. It is an ADE quiver gauge theory, with gauge group

n∏a=1

U(da), (2.11)

and Iab hypermultiplets in the bi-fundamental (da, db) representation for each

pair of nodes a and b. The theory has N = 2 supersymmetry in four dimen-

sions, since D5 branes break half the supersymmetry of IIB on X. The rank

da of the gauge group associated to the a-th node of the quiver is the number

of D5 branes wrapping the 2-cycle Sa. The hypermultiplets come from the

intersections of cycles Sa with Sb. This follows from a computation we can

4The 1/g2YM in five dimensions has units of mass. The τ is the dimensionless com-bination τ ∼ 1/(g2YMms). The gauge theory description is applicable for energies E/ms

less than 1, and the theory is weakly coupled for E/ms less than τ . When we study thepartition function, exp(−τ) will be the instanton expansion parameter, and we will wantthis to be less than 1, so we only need Re(τ) > 0.

10

do locally, near an intersection point. A non-zero intersection number Iab

of Sa with Sb, for distinct a and b, means that they intersect transversally

at Iab points. At a transversal intersection of 2 holomorphic 2-cycles in a 4-

manifold, there are 4 directions in which open strings with endpoints on the

branes have DN boundary conditions, leading to a massless bi-fundamental

hypermultiplet. The U(1) gauge groups of the D5 branes wrapping cycles are

actually massive, by Green-Schwarz mechanism [12], so the gauge groups are

SU(da) not U(da). Correspondingly, the Coulomb moduli associated with

the U(1) centers are parameters of the theory, not moduli. Nevertheless, the

effects of these U(1)’s remain: for example, due to stringy effects [42], the

partition function is that of a U(da) theory. For this reason, we will write

the gauge group with U(1) factors included, trusting the reader can keep in

mind the subtle point. (The issue of the U(1)’s was discussed in [13, 14],

from a related perspective.) The D5 branes on S∗ do not contribute to the

gauge group, since the cycle is non-compact, but they do contribute matter

fields: The intersections of non-compact cycles with compact cycles lead to

additional fundamental matter hypermultiplets. Since S∗a correspond to fun-

damental weights wa, they do not intersect Sb for b 6= a, see (2.4). Thus,

with S∗ as in (2.5), there are ma fundamental hypermultiplets on the a’th

node. In section 6, we will work out examples of 5d quiver gauge theories

that describe the corresponding little string theory on a sphere with three

full punctures. The resulting quivers are given in figures 2-6. In the An case,

this corresponds to the so called TN theory, with N = n+ 1; this quiver was

obtained earlier in [1] and studied further in [13, 14]. The rest are new.

While the theory has the super-Poincare invariance of a 4d N = 2 theory,

it is a 5d N = 1 theory compactified on a circle of radius R. Recall that

D5 branes are points on C = R× S1(R), and we are keeping ms finite. The

zero modes of strings that wind around S1(R) lead to a Kaluza-Klein tower

11

Figure 2: 5d gauge theory describing An little string with 3 full punctures.

Figure 3: 5d gauge theory describing Dn little string with 3 full punctures.

of states on the T-dual circle of radius

R =1

m2sR. (2.12)

The resulting tower of states affects the low energy physics [43]. For example,

the supersymmetric partition function of the theory depends on R, as we

will review in section 2.2. (Another way to see this it to do T-duality on

the circle. This relates D5 branes which are points on S1(R) to D6 branes

wrapping S1(R). In the D6 brane description, the fact that the low energy

theory is a five dimensional theory on a circle of radius R is manifest.)

The moduli of the (2, 0) theory in six dimensions become parameters in

four dimensions. They determine the couplings of the D5 brane gauge theory.

The complex combinations of the moduli which we called τa =∫Sa

(m2s

gsωI +

iBRR) are the gauge couplings of the effective 4d gauge theory on the D5

branes. The triplets of N = 2 Fayet-Iliopolous parameters, one for each

node of the quiver, come from the remaining 6d moduli,∫Sam2s ωJ,K/gs and

12

Figure 4: 5d gauge theory describing E6 little string with 3 full punctures.

Figure 5: 5d gauge theory describing E7 little string with 3 full punctures.

Figure 6: 5d gauge theory describing E8 little string with 3 full punctures.

∫SaBNS/gs. The only other parameters in the theory are the masses of the

fundamental hypermultiplets. These come from the positions of the non-

compact D5 branes on C: the non-compactness of the cycles in X renders

these non-dynamical as well. Finally, since the U(1) centers of the gauge

group are not dynamical, the Coulomb moduli associated with them are

13

parameters of the theory as well.

2.1.4 (2, 0) CFT From The Little String

The (2, 0) little string theory on C is not conformal. To recover the (2, 0)

6d CFT theory on C we need to take the string scale ms to infinity5, while

keeping the Riemann surface C and the moduli of the (2, 0) theory in (2.2)

fixed. Furthermore, we will send to zero ∆xms, where ∆x are the relative

positions of D5 branes on C.The gauge theory on D5 branes becomes four-dimensional since the ra-

dius R = 1/(m2sR) of the 5d circle vanishes in the limit. The Lagrangian

description of it from the previous section breaks down: the inverse gauge

coupling τa of the 4d theory vanishes since m2sτ is one of the moduli of the

(2, 0) SCFT, and needs to stay fixed in the limit. There is no energy scale

where the ADE quiver gauge theory description is weakly coupled. As we

will see, for g 6= An, the effective rank of the gauge group becomes smaller in

the limit. Nevertheless, we can learn a lot about the CFT by working with

the mass-deformed theory, and taking the massless limit only at the very

end. As is commonly the case, the massive theory is easier to understand.

Finally, note that there is another way to take the R to zero limit, where

we keep τ ’s finite. This would result in a 4d conformal field theory with the

same quiver as the 5d theory, studied in [19, 20]. This does not describe the

(2, 0) theory on C, as the moduli of the (2, 0) theory, proportional to τam2s,

go to infinity in the field space.

5More precisely, we are considering energies E where E/ms goes to zero, and ER isfinite. In the limit, ER = E/(m2

sR) goes to zero, so the theory becomes four dimensional.Keeping the (2, 0) modulus φ fixed means φ/E2 is finite in the limit. If φ = τm2

s, whereτ is the D5 gauge coupling, then τ must go to zero. This is the strong coupling limit.

14

2.2 Partition Function of Little String Theory on C

We can use the gauge theory description from section 2.1.3 to compute the

supersymmetric partition function of the (2, 0) little string theory on C ×C2

with an arbitrary collection of defects at points of C:

ZLittle String(C × C2) = Z5d (S1 × C2)

The supersymmetric partition function should be an RG invariant, so the

description from section 2.1.3 is valid at long distances and should suffice for

non-zero τa’s. More precisely, we will replace C2 with the four dimensional

Ω-background [17, 18] as a regulator, and then the partition function is the

trace

Z5d (S1 × C2) = Tr(−1)F g.

Here g = qS1−SRt−S2+SR ; S1, S2 are generators of the rotations around the

two complex planes in C2, F is the fermion number, and q = eRε1 , t = e−Rε2 .

To form a supersymmetric trace, we make use of the SU(2)R R-symmetry

that is preserved by the configuration of D5 branes (this is the subgroup of

the SO(5)R R-symmetry of IIB string theory on X which acts by rotating

triplets of scalars in (2.9) that vanish in the brane background). We will make

use only of the U(1)R ⊂ SU(2)R subgroup, generated by SR, and which acts

by rotations of the scalars coming from ωK and BNS in (2.9). The trace is

the trace in going around the 5d circle. The gauge theory partition function,

in addition to q and t, depends on τ ’s, the 5d gauge couplings, which are the

moduli of the (2, 0) theory in 6d, see (2.10). It depends on the masses of the

fundamental hypermultiplets - these are the positions of the non-compact

D5 branes on C. Finally, it depends on the Coulomb moduli; these are the

positions of compact D5 branes on C.The partition function of quiver gauge theories of this type was computed

in [19], using equivariant integration on the instanton moduli space. One

15

works equivariantly with respect to all the U(1) symmetries in the problem:

the U(1) symmetries coming from the Cartan subalgebra of the D5 gauge

and flavor groups, and the rotations S1, S2, and SR. The partition function

becomes a sum over fixed points in instanton moduli spaces. The latter are

labeled by a collection R of 2d partitions:

R = R(a),ia=1,...n; i=1,...da ,

one for each U(1) factor in the gauge group on the D5 branes. The 2d parti-

tions describe how instantons of the corresponding U(1) factor get ”stacked”

at the fixed point in C2. At each node, there are as many 2d Young diagrams

as the rank of the corresponding unitary gauge group. The contribution of

each fixed point to the partition function,

Z5d = r5d

∑R

I5d,R(q, t; a,m, τ), (2.13)

is a product of factors

I5d,R = eτ ·R ·n∏a=1

zVa, ~R(a) zHa, ~R(a) zCS,~R(a) ·n∏

a,b=1

zHa,b, ~R(a), ~R(b) , (2.14)

which we can read off from the D5 brane quiver (we follow the conventions

of [44]). The gauge group on the a-th node of the ADE quiver is U(da). The

corresponding vector multiplets contribute

zVa, ~R(a) =∏

1≤I,J≤da

[NR(a),iR(a),j(e(a),I/e(a),J)]−1.

Here, e(b),I = exp(Ra(b),I) encode the da Coulomb branch parameters of the

U(da) gauge group. There are ma hypermultiplets charged in fundamental

representation of the U(da) gauge group. They contribute to the partition

function as:

16

zHa, ~R(a) =∏

1≤α≤ma

∏1≤I≤da

N∅R(a),I(vf(a),α/e(a),I).

The masses of the hypermultiplets are encoded in f(a),α = exp(Rm(a),i), where

α takes ma values. In what follows, we write v = (q/t)1/2. For every pair of

nodes a, b connected by an edge in the Dynkin diagram, we get a bifunda-

mental hypermultiplet. Its contribution to the partition function is:

zHa,b, ~R(a), ~R(b) =∏

1≤i≤da

∏1≤j≤db

[NR(a),IR(b),J(e(a),I/e(b),J)]Ia,b .

where Ia,b is the incidence matrix, equal to either 1 or 0, depending on

whether, in the Dynkin graph, there is an arrow starting on the a’th node and

ending on the b’th one or not. The contribution of 5d N = 1 Chern-Simons

terms kCSa for this node reads

zCS,~R(a) =∏

1≤I≤da

(TR(a),I

)kCSaHere, TR is defined as TR = (−1)|R|q‖R‖/2t−‖R

t‖/2. The 5d N = 1 Chern-

Simons terms can be determined by conformal invariance; with the rest of

the partition function as written, kCSa on the a-the node is the difference of

the ranks of the gauge group on that node, and the following node(s). The

gauge couplings keep track of the total instanton charge, via the combination

τ ·R =n∑a=1

da∑I=1

τa |R(a),I |.

The vector and hypermultiplet contributions are all given in terms of the

Nekrasov function NRP (Q), which is defined as:

NRP (Q) =∞∏i=1

∞∏j=1

ϕq(QqRi−Pj tj−i+1

)ϕq(QqRi−Pj tj−i

) ϕq(Qtj−i

)ϕq(Qtj−i+1

) . (2.15)

17

where ϕq(x) =∞∏n=0

(1− qnx) is the quantum dilogarithm. The normalization

factor r5d in (2.13) contains the tree level and the one loop contributions to

the partition function.

2.3 Integrable Systems, Bogomolny and Hitchin Equa-tions

The integrable system associated to the (2, 0) little string theory on C has

two descriptions. The first is in terms of the moduli space of G monopoles

on R × T 2. The second is in terms of a Hitchin-type system on C. These

integrable systems and their relation to 5d quiver gauge theories were studied

recently in [19, 20, 45], so we can focus here on the new aspect, namely, the

relation to the (2, 0) little string theory. The connection to integrable systems

emerges upon compactifying the theory on an additional circle, which we

take to have the radius R′, so we study (2, 0) little string on C ×S1(R′), with

defects at points on C, as before.

2.3.1 Monopoles on R× T 2

The relation to monopoles on R× T 2 emerges upon T-duality on S1(R′). T-

duality relates IIB to type IIA string on X×C×S1(R′), where R′ = 1/(R′m2s).

It also relates D5 branes to D4 branes and (2, 0) little string on C×S1(R′) to

(1, 1) little string on C × S1(R′), since it remains a symmetry of the theory

as long as ms is finite. The (1, 1) little string becomes, at low energies, the

maximally supersymmetric gauge theory in 6d, with gauge group based on

the Lie algebra g. The D4 branes are g monopoles: they are points on

C×S1(R′), magnetically charged under the gauge fields of the 6d little string

(the gauge fields originate from periods of the RR 3-form potential on the

2-cycles in X). The identification of D4 branes with monopoles identifies the

Coulomb branch of the D5 brane gauge theory on S1(R′) with the moduli

18

space of g-monopoles on C×S1(R′); the latter is a hyperkahler manifoldMC

of quaternionic dimension∑n

a=1 da.

D-Branes as Monopoles

The D4 branes wrapping the compact 2-cycles are non-abelian monopoles.

In gauge theory, the non-abelian monopole charges are valued [19] in the co-

root lattice Λ∨ of g. The charges of D4 branes wrapping compact 2-cycles

live in H2cmpt(X,Z), corresponding to compactly supported cohomology. This

group is indeed the same as the co-root lattice, Λ∨ = H2cmpt(X,Z) [37]. Using

Poincare duality, or the inner product on the Lie algebra, we can identify this

lattice with the homology of compact support H2(X,Z) = Λ, or equivalently,

with the root lattice. This identifies the charge of the D4 brane wrapping

the cycle S with the homology class of the cycle [S] =∑n

a=1 daea itself.

The D4 branes wrapping non-compact cycles, are singular, Dirac monopoles

[? ? ]. In gauge theory, the charges of Dirac monopoles are supported in

the co-weight lattice Λ∨∗ of g. The charges of non-compact branes are in

H2(X,Z), containing cohomology of both compact and non-compact sup-

port. The cohomology group H2(X,Z) = Λ∨∗ is the co-weight lattice, as it

is dual to the root lattice H2(X,Z) = Λ. Poincare duality, in turn, pro-

vides identification between H2(X,Z) with H2(X, ∂X;Z), so identifies the

co-weight and weight lattices, where the identification becomes the identity

map for a simply laced g. The charge of the Dirac monopole corresponding

to D4 branes wrapping the non-compact cycle S∗ equals [S∗] = −∑

amawa,

the class of the cycle in (2.5).

Monopoles are solutions to Bogomolny equations:

Dφ = ∗F. (2.16)

Here F is the curvature of the gauge field, and φ is the real scalar field which

19

approaches a constant value at infinity; both are valued in the Lie algebra

g. In our case F and φ come from the (1, 1) little string. The scalar φ

is φa =∫Sam3s ω

I/g′s, where g′s is the IIA string coupling, related to the IIB

string coupling by ms/g′s = R′m2

s/gs. We need to solve Bogomolny equations

on R× T 2 = C × S1(R′).

2.3.2 Hitchin-Type System on C

If R′ is small, we can forget about the positions of the monopoles on the circle

S1(R′). This corresponds to going back to the original D5 brane description

on a large circle, of radius R′. Then, the Bogomolny equations on C ×S1(R′)

reduce to Hitchin-type equations on C,

F = [ϕ, ϕ]

Dxϕ = 0 = Dxϕ.(2.17)

These are not exactly the equations considered by Hitchin, because the imag-

inary part of ϕ is a periodic scalar, with period 1/R′. In going from D4 branes

back to D5 branes, φ in (2.16) was complexified to ϕ = φ + iAθ, where Aθ

is the holonomy of the (1, 1) gauge field around the S1(R′) circle. The peri-

odicity of the Higgs field ϕ reflects the fact we study 5d theory on a circle,

as opposed to a purely four dimensional one. Given a solution for ϕ(x) of

(2.17) corresponding to D5 brane defects, we can write the Seiberg-Witten

curve as the spectral curve of ϕ, taken in some representation Q of g:

detQ(eR′p − eR′ϕ(x)) = 0. (2.18)

The factor of the radius R′ accompanying ϕ in the formula is determined

so that R′ϕ has period 2πi, and the exponent is single valued. One can

phrase this as studying a group-valued Hitchin system on C, see [45]. (In the

language of IIB on X, the periodic direction comes from the period of the

RR B-field on 2-cycles of ALE. It is related to the holonomy of the bulk (1, 1)

20

6d gauge field on the circle: R′Aaθ =∫SaBRR.) We can take the determinant

in any representation Q of the group.

When we send ms to infinity, the (2, 0) little string theory becomes the

(2, 0) 6d CFT. We want to do this while keeping the scalar fields of the (2, 0)

theory fixed, and also the radii of the two circles R and R′ it is compactified

on. Then the radius R′ goes to zero, and the 6d (1, 1) little string theory

compactified on C × S1(R′) becomes 5d, maximally supersymmetric Yang-

Mills on C, with inverse gauge coupling squared equal to m2sR′ = 1/R′. At

the same time, the periodic direction in ϕ decompactifies, and we recover the

ordinary Hitchin system on C, associated to the Lie algebra g. This is the

integrable system associated to class S theory on C as in [4]. In particular, in

the limit, we will recover the spectral curve of the Hitchin integrable system:

detQ(p− ϕ(x)) = 0. (2.19)

2.4 The Weight System WS

It turns out to be very fruitful to study the theory on the Higgs branch,

where the gauge group∏n

a=1 U(da) is broken to its U(1) centers, one for

each node. We force the theory onto the Higgs branch by turning on the

remaining moduli of the (2, 0) theory∫SaωJ,K ,

∫SaBNS (see [46] for a detailed

analysis from gauge theory perspective); these are the FI parameters in the

D5 brane gauge theory (2.9). The deformation is normalizable, affecting only

the geometry of X near the singularity.

On the Higgs branch, the compact and non-compact D5 branes must

recombine: the deformation changes the supersymmetries preserved by the

compact D5 branes (it changes their central charges via (2.9)), but not the

supersymmetries preserved by the non-compact ones (these can be detected

at infinity inX). As a consequence, the branes on S in (2.6) and on S∗ in (2.5)

are no longer mutually supersymmetric. From the monopole perspective,

21

this is the statement that once the gauge group is Higgsed, there are no non-

abelian monopole solutions to (2.16). Instead, all monopoles recombine into

Dirac monopoles of the leftover, abelian gauge group. Correspondingly, after

Higgsing of the gauge group, the D5 branes wrapping S and S∗ re-combine

to form D5 branes wrapping a collection of non-compact cycle S∗i , whose

homology classes are elements ωi of the weight lattice Λ∗ = H2(X, ∂X,Z):

ωi = [S∗i ] ∈ Λ∗. (2.20)

The classes ωi can be determined as follows. Each of ωi’s comes from one of

the non-compact D5 branes on S∗. For the branes to bind, the positions of

compact branes must coincide with positions of one of the non-compact D5

branes on C. The positions of non-compact D5 branes are mass parameters

of the quiver gauge theory, the positions of compact D5 branes on C are

Coulomb moduli; when a Coulomb modulus coincides with one of the masses,

the corresponding fundamental hypermultiplet becomes massless and can get

expectation values. This, in turn, describes the D5 branes binding (see [47]

for a similar example), and allows supersymmetry to be preserved in presence

of non-zero FI terms. Thus, ωi has the form −wa plus the sum of positive

simple roots ea, from bound compact branes. Not any such combination will

correspond to truly bound branes: a sufficient condition is that ωi is in the

Weyl orbit of −wa = [S∗a]: this means that there is a point in the moduli

space of the theory on X where S∗i and S∗a look the same. Furthermore, the

collection of weights

WS = ωii (2.21)

we get must be such that it accounts for all the D5 brane charge in [S∗] and

in [S]. One simple consequence is that the number of ωi’s is thus the rank

of the 5d flavor group,∑n

a=1ma. The fact that the net D5 charge is zero

22

[S + S∗] = 0 implies that ∑ωi∈WS

ωi = 0, (2.22)

which is thus equivalent to (2.8).

Figure 7: D5 branes in classes ea and −wa bind to a brane in class −wa + ea.

In section 6, we will work out some simple examples. The most canonical

type of defect uses the fact that the weight lattice of a Lie algebra of rank

n is n dimensional. We can construct a set WS by picking any n+ 1 weight

vectors which lie in weyl-group orbit of the fundamental weights −wa, for

some a, and which sum up to zero, and such that n of them span Λ∗. This

leads to a a full puncture defect of the (2, 0) little string on C, and the

corresponding gauge theory description in figures 2-6. We can also string

sets of such defects, for a k punctured sphere.

The S-curve from Dirac Monopoles and WS

The Seiberg-Witten curve (2.18),(2.19), captures the low energy physics

of the theory on the Coulomb branch and coincides with the spectral curve

of the corresponding integrable system. The Seiberg-Witten curves for this

class of theories were found in [19, 48], using a fairly involved analysis. There

is a simple way to obtain the Seiberg-Witten curve: it can be derived from

another curve, called the S-curve in [1]. The S-curve is the Seiberg-Witten

23

curve at the point where Higgs and the Coulomb branches meet.

We know the Seiberg-Witten curve if we know the Higgs field ϕ(x) solving

(2.17), or equivalently (2.16). On the Higgs branch, when all the monopoles

become Dirac monopoles, solving the Bogomolny equations in (2.16) explic-

itly is easy, and the S curve follows. The effect on ϕ of adding a Dirac

monopole of charge ω∨i , at a point x = R′mi on C, is to shift:

eR′ϕ → eR

′ϕ · (1− ex−R′mi)−w∨i . (2.23)

This follows [19] by solving Bogomolny equations for φ on C×S1(R′), and then

dropping the dependence on positions on S1. (In [19], one had monopoles on

a plane parameterized by x. We compactify x into a cylinder C by adding

infinitely many images to impose x ∼ x+ 2πi. This amounts to replacing x

by ex.) In the absence of Dirac monopoles, the ϕ would have been constant,

given by6 the vacuum value R′ϕ = τ . (As vectors, we can simply identify

the co-weights and weights, since g is simply laced. At times, we will want

to keep the distinction in the labeling in order to recall that the Higgs field

eϕS(z) lives in the maximal torus of g.) Thus, the Higgs field ϕ = ϕS(x)

solving the Hitchin equation at the point where the Higgs and the Coulomb

branches meet is

eR′ϕS(x) = eτ

∏ωVi ∈WS

(1− ex−R′mi)−ω∨i . (2.24)

The S curve in representation Q is simply obtained by specializing ϕ =

ϕS(x) in (2.18),(2.19). This amounts to finding the eigenvalues of ϕS in

representation Q and computing the determinant by taking the product.

From the S-curve, we can recover the Seiberg-Witten curve at a generic

point on the Coulomb branch by turning on generic normalizable moduli.

This was discovered and explained in [29–32, 49].

6Recall that, in IIB variables, ϕa = (ea, ϕ) = R′m2s

∫Sa

(m2sωI/gs + iBRR) = τa/R

′.

24

2.4.1 Defects in (2, 0) CFT and Little String

In the limit in which (2, 0) little string becomes (2, 0) CFT, we expect to

get an ordinary Hitchin system on C, with defects. The Higgs field ϕS cor-

responding to it is obtained from the solution in (2.26) by taking a limit

described in section 2.1.4. We take ms to infinity, keeping ϕ, as well as R

and R′, fixed in the limit.7 This means that we need to take τ to zero in the

limit, since it equals to R′ times the value of ϕ in the absence of monopoles.

We will denote by α0 the vacuum value of ϕ, α0 = τ/R′. A more subtle

aspect of the limit is that we need to collide the fivebranes, by taking R′ to

zero and keeping ∆x/R′, their relative distances on C, fixed. We can write

the position xi of a D5 brane from ωi as

exi = zP eR′αi,P , (2.25)

and keep zP and α fixed as we take R′ to zero. Not all the D5 branes need

to coalesce at the same point on C; instead, we may group them into subsets

of weights in WP that separately sum to zero∑ωi∈WP

ωi = 0.

Because the sum of the wi’s in WP vanishes, there are no redundant param-

eters in (2.25); shifting all αi’s by the same amount does not affect ϕ. It is

easy to see that, in the limit where ms goes to infinity, we get:

ϕS(x) = α0 +∑P

∑ωi∈WP

αi,P ω∨i

zPe−x − 1.

It is more convenient to rewrite this in terms of a new variable z = ex. Since

ϕ(x)dx is a one form on the Riemann surface, it the change of variables

7The scalar ϕ(x) equals, up to a factor of R′, the modulus of the (2, 0) theory, ϕ2,0 =m2

s

∫Sa

(m2sωI + iBRR).

25

transforms it to ϕ(z) = ϕdx/dz = ϕ(z)/z, and consequently,

ϕS(z) =α0

z+∑P

∑ωi∈WP

αi,P ω∨i

zP − z. (2.26)

This tells us that, in the (2, 0) CFT, we have poles on C at z = zP , with

residues

αP =∑

ωi∈WP

αi,P ω∨i .

This is the expected behavior of the Higgs field ϕ near the defects in the

(2, 0) theory on C [3]. As we will discuss in section 6, when we take WP to

correspond to a set of n+1 weights ωi that span the weight lattice, the residue

αP is generic, which leads to a full puncture at z = zP . It would be clearly

important to connect the description of defects in the (2, 0) theory which we

derived here, to the a-priori different description of defects in [40, 41].

2.5 Coulomb Branch for Finite and Infinite ms

At generic points on the Coulomb branch, ϕ(x) is no longer diagonalizable.

The Seiberg-Witten curve is no longer the same as the S-curve, only their

asymptotic behavior, near the punctures on C, is the same. This is the case

since Coulomb branch moduli are normalizable deformations of the Seiberg-

Witten curve. The dimension of the Coulomb branch of little string theory

is∑n

a=1(da − 1), where da are the ranks of the gauge groups in (2.11). This

takes into the account that, while the Coulomb moduli coming from the U(1)

centers of the 5d gauge groups are not normalizable – they affect residues of

ϕ at the puncture at z =∞.

The dimension of the Coulomb branch of the (2, 0) CFT compactified on

C is generically smaller – in the ms to zero limit, some deformations that were

distinct at finite ms become indistinguishable. One way to count the moduli

in the limit is from the perspective of IIB on X×C. In the ms to infinity limit,

when the moduli space becomes (R)5/W , we can use an R symmetry rotation

26

to reinterpret ϕa =∫Sam4sωI/gs + im2

sBRR as ϕa =∫Sam4s(ωJ + iωK)/gs.

This does not change the theory in the limit, but the latter has a different

geometric interpretation: it is a complex structure deformation of X × C,which turns it into a Calabi-Yau threefold Y , since ϕ(x) varies over C. This

lets one use techniques of complex geometry to count the number of Coulomb

moduli as the number of normalizable complex structure deformations of Y .

For the resulting class of Calabi-Yau manifolds, the counting was done in

[32, 49]. For example, if one assumes that C starts out as a sphere with k

full punctures, where the residues αP of ϕ(z) are all generic, the dimension

of the Coulomb branch is (k − 2)h+(g)− n, the where h+(g) is the number

of positive roots of g. This is not surprising, as we are embedding the (2, 0)

CFT into a theory with a lot more degrees of freedom.

3 ADE Little String and D3 Branes

On the Higgs branch of little string theory, the bulk theory is abelianized,

the D5 branes are all non-compact. At the same time, there is a new class of

brane that plays an important role: these are D3 branes which are at points

on C and which wrap compact 2-cycles in X. (The D3 branes wrapping non-

compact 2-cycles are also important; they are codimension 4 defects of the

little string, studied in [36].) The D3 branes survive the little string limit,

for the same reason D5 branes in section 2 did; their tensions remain finite.

3.1 D3 Brane Gauge Theory

String theory provides a derivation of the gauge theory on the D3 branes

wrapping compact 2-cycles in X in presence of non-compact D5 branes on

cycles WS . Let the homology class of the D3 branes in H2(X,Z) = Λ be

[D] =n∑a=1

Na ea ∈ Λ (3.1)

27

where Na are positive integers. In the absence of D5 branes on X, the theory

on the D3 branes was derived by Douglas and Moore in the seminal paper

[12]. The theory is an ADE quiver theory with N = 4 supersymmetry in 3d,

with U(Na) gauge group for the a-th node of the quiver, leading to

n∏a=1

U(Na), (3.2)

and Iab bufundamental matter hypermultiplets for each pair (a, b) of nodes

of the Dynkin diagram. From the N = 2 perspective, each vector multiplet

contains an adjoint chiral multiplet, and there is a cubic superpotential.

It remains to deduce the effect of the D5 branes. Recall that, on the Higgs

branch, the D5 branes wrap a collection of cycles S∗i , whose homology

classes make up WS . Quantizing D3-D5 strings, we get chiral or anti-chiral

multiplet of N = 2 supersymmetry for each intersection point in X between

the compact 2-cycle Sa wrapped by the D3 branes and S∗i wrapped by the D5

branes. This follows since there are 6 Dirichlet-Neumann directions for open

strings with one boundary on D5 branes and one on D3 branes: two come

from the R4 part; four more DN directions come from the fact that the branes

wrap two-cycles in X, intersecting transversally. The matter fields preserve 4

supersymmetries, since the D5 branes break half of the supersymmetry of X.

This requires knowing the details of the geometry of the cycles wrapped by

the branes. A much simpler quantity to determine is the net number of anti-

chiral minus the chiral multiplets transforming in fundamental representation

of the gauge group [49]:

#(Sa, S∗i ) = (ea, ωi) (3.3)

for which we only need to know the homology classes [Sa] = ea of the compact

and the non-compact cycles [S∗i ] = ωi (the anti-chiral multiplets are the

CPT conjugates of the chiral ones). In the examples that are relevant for

28

us, the geometry of the cycles S∗i is simple, and the index (3.3) turns out

to detects the full chiral matter content of the theory. Finally, the fact that

the theory on D3 branes is really a 3d N = 2 gauge theory on a circle of

radius R follows most easily from T-duality that maps D3 branes at points

on C = R× S1(R) to D4 branes wrapping S1(R). We chose the moduli of X

so that∫SamsωJ/gs > 0 in (2.2) and set the gauge couplings of the theory

to zero on the D3 branes. The parameters τa in (2.10) are the real FI terms,

which are complexified because the theory is a 3d gauge theory on a circle.

The remaining moduli in (2.2) are the complex FI terms of the D3 brane

gauge theory, which we set to zero.

3.2 D3 Branes are Vortices

The D3 branes realize vortices in the D5 brane gauge theory. Vortices are

co-dimension two solutions of gauge theories on the Higgs branch, where the

vortex charge is the magnetic flux in two directions traverse to the vortex. A

generic collection of vortices in 5d N = 1 gauge theories are BPS if the 5d FI

parameters are aligned. At each node, the triplet of FI terms transform as a

vector under the SU(2)R symmetry rotations. The orientation of this vector

determines the supersymmetry preserved by the vortex. In our setting, the

5d FI parameters are the moduli of the little string in (2.9). The background

we consider has∫Sam2sωJ/gs > 0 as the only non-zero FI terms in (2.9). The

vortices are in fact the supersymmetric vacua of the theory on the D3 branes.

Due to non-zero 3d FI terms (recall that Re(τa) > 0) in a supersymmetric

vacuum, the chiral multiplets from the D3-D5 strings need to get expectation

values. This describes D3 branes ending on the D5 branes. As is well known,

this turns on magnetic flux on the D5 brane, transverse to the D3 branes

[50], consistent with the vortex interpretation.

The Higgs branch of the theory on the D3 branes, from the previous

29

section, is the moduli space of vortices. We have thus derived, from string

theory, the description of the moduli spaces of vortices in a large class of

ADE quiver N = 2 theories. As far as we are aware, the result is novel,

except in some special cases. (In mathematical literature, the moduli space

of vortices is called the moduli space of quasi-maps; see for example [51],

where the quiver in figure 8 appeared before, precisely for this purpose.)

3.3 Partition Function of D3 Branes

Consider now the partition function of the D3 brane theory in Ω-background.

From the perspective of the little string, the partition function corresponds

to the theory in exactly the same background as in section 2.2, except now

the bulk theory is on the Higgs branch, and all the D5 branes are non-

compact. In addition we have Na D3 branes wrapping the 2-cycle Sa and

the complex plane rotated with parameter q; we take the plane rotated by t

to be transverse to the branes. Of course, for the same reason the theory on

the D5 branes wrapping 2-cycles in X was a five dimensional N = 1 gauge

theory, the theory on the D3 branes on 2-cycles is a three dimensional gauge

theory with N = 2 supersymmetry, both on a circle of radius S1(R).

The partition function of the theory Z3d is

Z3d = Tr(−1)F g, (3.4)

where as before, g = qS1−SRt−S2+SR is a combination of SO(2) rotations S1,2

of the two copies of C. SR is the same R-symmetry rotation as in section 2.2;

we chose the background in section 3.1 so that the symmetry is preserved.

Before we add D5 branes, the theory has N = 4 supersymmetry. Then,

the generators we call SR and S2 are the generators of the U(1)H × U(1)V

subgroup of the SU(2)H × SU(2)V R-symmetries that act on the Higgs and

the Coulomb branches of the theory, respectively. This identification comes

from the fact that S2 acts by phase rotation of the N = 2 chiral adjoint

30

multiplet which sits inside the N = 4 vector multiplet, and does not act

on the hypermultiplets. The bifundamental hypermultiplet transforms as a

doublet of SU(2)H . From the perspective of the 3d gauge theory, if we take

S1 to rotate the D3 brane world-volume, and S2 to rotate the space transverse

to brane, then both S2 and SR are R-symmetries. Since the theory on the D3

branes has 3d N = 2 supersymmetry that has at most a U(1)R symmetry,

the difference SR − S2 is in fact a global symmetry.

The partition function can be computed as the integral over the Coulomb

branch:

Z3d =

∫dx I3d(x). (3.5)

The x’s are the Coulomb moduli of the 3d gauge theory, the positions on

C of the D3 branes. The integrand I3d(x) is the contribution to the index

(3.4) from one loop integrating out of massive charged matter fields (detailed

study of partition functions of this type is in [52–56]), together with classical

terms. It can be read off from the quiver, by including contributions of all

gauge and matter fields

I3d(x) = r3d

n∏a=1

I3da (xa) · I3d

a,V (xa, f) ·∏a<b

I3dab (xa, xb) (3.6)

The contribution of the N = 4 U(Na) vector multiplets is

I3da (xa) = e

∑NaI=1 τaxa,I

∏1≤I 6=J≤N(a)

ϕq(ex(a)I −x

(a)J )

ϕq(t ex(a)I −x

(a)J )

, (3.7)

where the numerator comes from W -bosons, the denominator from the ad-

joint chiral multiplet within the vector multiplet. The bifundamental hyper-

multiplets, corresponding to a pair of nodes a and b, give:

I3dab (xa, xb) =

∏1≤I≤N(a)

∏1≤J≤N(b)

(ϕq(v t ex(a)I −x(b)J )

ϕq(v ex(a)I −x

(b)J )

)Iab, (3.8)

31

The contribution is non-trivial only for the pairs of nodes connected by a

link in the Dynkin diagram. For every ωi ∈ WS , we get a collection of chi-

ral multiplets and anti-chiral multiplets. A chiral multiplet in fundamental

representation of the gauge group on the a-th node, with SR R-charge −r/2contributes

∏1≤I≤Na

(ϕq(v

rf/ex(a)I ))−1

, while an anti-chiral multiplet in fun-

damental representation contributes∏

1≤I≤Na

(ϕq(v

rf/ex(a)I ))

, where f is the

flavor fugacity and v =√q/t. The function ϕq(z) is as in section 2.2. For

each node, we get a contribution of the form

I3da,V (xa, f) =

∏wi∈WS

I3da,ωi

(xa, fi)

where

I3da,ωi

(xa, fi) (3.9)

captures the contributions of all the chiral and anti-chiral matter fields com-

ing from strings stretching between the D5 brane wrapping [S∗i ] = ωi and

the D3 brane on ea = [Sa]. This depends on fi = exp (Rmi), encoding the

position xi = Rmi of the D5 brane on C. To write down the explicit formula

in (3.9), one in general needs to know the spectrum, and not just the index

(3.3). The N = 4 matter contribution in (3.8), by contrast, is fixed by su-

persymmetry. For the theories in Fig.1 we will give the explicit formulas in

section 6. The integral runs over the Coulomb branch moduli for each of the

n gauge group factors in (3.2):

”

∫dx ” =

1

|WG3d|

n∏a=1

∫dNax(a). (3.10)

We still need to specify the integration cycle in (3.5). The integration cycles,

in turn, correspond to vacua of the 3d gauge theory, see for example [52].

We will postpone discussing the contours until section 5, when we will need

them.

32

4 ADE Toda and its q-deformation

The D3 branes on X have a close relation to a 2d conformal field theory.

The partition function of the gauge theory on D3 branes in (3.5) turns out

to be equal to a certain canonical ”q-deformation” of the ADE Toda CFT

conformal block on C withWq,t(g) vertex algebra symmetry, found by Frenkel

and Reshetikhin in [24]. The relation between them is manifest, as soon as

one recalls basics of the Toda CFT, and the construction in [24]. Moreover,

taking the mS to infinity limit that brings (2, 0) little string back to (2, 0)

CFT, the q-deformed Toda CFT reduces to the ordinary Toda CFT, and

Wq,t(g) to the W(g) algebra. However, just as was the case for the D5

branes, the limit does not correspond to a partition function of any gauge

theory with a Lagrangian – the relation between the two is simple only within

the little string theory.

4.1 Free Field Toda CFT

The ADE Toda field theory can be written in terms of n = rk(g) free bosons

in two dimensions with a background charge contribution and the Toda po-

tential that couples them:

SToda =

∫dzdz

√g gzz[(∂zϕ, ∂zϕ) + (ρ, ϕ)QR +

n∑a=1

e(ea,ϕ)/b]. (4.1)

The field ϕ is a vector in the n-dimensional (co-)weight space, (, ) is the

Killing form on the Cartan subalgebra of g, ρ is the Weyl vector, and Q =

b+ 1/b. As before, ea label the simple positive roots. The Toda CFT has an

extended conformal symmetry, a W(g) algebra symmetry. (For a review of

W algebras see [57]. The free field formalism for Toda CFT was discovered

in [58] and studied in the present context in [59–63]). The primary vertex

operators of the W(g) algebra are labeled by an n-dimensional vector of

33

momenta α, and given by:

Vα(z) = e(α,ϕ(z)). (4.2)

The free field conformal blocks of the Toda CFT have a particularly simple

form:

〈Vα1(z1) . . . Vαk(zk)n∏a=1

QNaa 〉free (4.3)

where screening charges

Q(a) =

∮dxSa(x)

are the integrals over the screening current operators Sa(x), one for each

simple root,

S(a)(z) = e(ea,φ(z))/b. (4.4)

We will only give a rough sketch of the derivation of (4.3) (see [64] for de-

tails). Consider treating the Toda potential as a perturbation, expanding

and bringing down powers of the Toda potential. Each term is now a com-

putation of the correlation in a free field theory, with insertions of screening

charge integrals coming from Toda potential. Momentum conservation picks

out a single term, the one with the net vertex operator momentum plus the

momenta of the screening charges,

k∑i=1

αi +n∑a=1

Naea/b = 2Q. (4.5)

The last term comes from the background charge on a sphere, induced by the

curvature term in (4.1). Picking out the chiral half of the correlation, results

in (4.3). A more precise derivation of the constraint results directly from the

path integral, by integrating over the zero modes of the bosons [64]. The

constraint (4.5) says that one of the momenta, say the momentum α∞ of the

vertex operator at z = ∞, is fixed in terms of momenta αi of all the other

vertex operators and numbers of screening charge integrals Na. Computing

34

the correlators using Wick contractions, the conformal block has the form of

an integral over the positions x of Na screening current insertions

ZToda =

∫dx IToda(x) (4.6)

where the integrand IToda, coming form (4.3) is a product over two-point

function of the screening currents with themselves, running over all pairs of

nodes of the Dynkin diagram, and two-point functions of screening currents

and vertex operators:

IToda(x) =n∏a=1

ITodaa (xa) · Ia,V (xa, z) ·∏a<b

ITodaab (xa, xb) (4.7)

The two-point functions of screening currents at a fixed node contribute

ITodaa =∏

1≤I 6=J≤N(a)

〈Sa(x(a)I ), Sa(x

(a)J )〉free. (4.8)

It will become apparent momentarily that, in the q-deformed theory, these

directly correspond to N = 4 U(Na) vector multiplet contributions coming

from U(Na) gauge theory at a-th node. The two-point functions of screening

currents between distinct nodes a and b contribute

ITodaab =∏

1≤I≤N(a)

∏1≤J≤N(b)

〈Sa(x(a)I ), Sb(x

(b)J )〉free. (4.9)

These correspond to bifundamental hypermultiplet contributions. Finally,

the two-point functions of screening currents at a given node with all the

vertex operators,

ITodaa,V =k∏i=1

∏1≤I≤N(a)

〈Sa(x(a)I ), Vαi(zi)〉free, (4.10)

will correspond to chiral matter contributions. The two-point functions are

those of ordinary free bosons, so they are simply equal8 to:

〈Sa(x), Sb(x′)〉free = (x− x′)b2Cab (4.11)

8We have been cavalier throughout with the momentum zero modes. It is a straight-forward exercise to restore these.

35

〈Sa(x), Vα(z)〉free = (x− z)(α,ea). (4.12)

The relation to a gauge theory arises only after q-deformation. The structure

of the partition function as an integral in (4.6), (4.7), is reminiscent of the 3d

gauge theory partition function in (3.5), (3.6). After we q-deform the Toda

CFT, they become the same.

Contours and Fusion Multiplicities

To fully specify the conformal block, we need to make a choice of con-

tour in (4.6). Conformal blocks can be obtained as solutions to differential

equations of hypergeometric type, and (4.6) can be viewed as providing an

integral solution to the equation, see for example [64, 65]. Generically, there

is a finite dimensional space of solutions to such equations, and choosing a

contour picks out a specific one. We will compute the dimension of the space

of conformal blocks of W(g) algebra, following [38, 64, 66], where the calcu-

lation was done for g = An. The W(g) algebra is a vertex operator algebra

with n generators W(sa), labeled by their spins sa. It has Virasoro algebra

as a subalgebra, generated by the spin 2 generator, the stress tensor. The

vertex operators Vα(z) in (4.2) are primaries of the entire W algebra, not

just of its Virasoro subalgebra. The Virasoro symmetry suffices to determine

the correlation functions of all the Virasoro descendants in terms of those

of the Virasoro primaries. But, since W algebra is bigger than the Virasoro

algebra, the set of Virasoro primaries is larger than the set of W algebra

primaries – it includes not only Vα(z), but also theirW algebra descendants.

One can use the W algebra and the Ward identities to reduce the space of

the descendant insertions. For the k-point function on a sphere, involving

k primary operators of generic momenta, the number of W algebra gener-

ators that cannot be removed is (k − 2)h+ − n [66], where h+ = h+(g) is

the number of positive roots of g. This is the additional data one needs to

36

specify besides the k external momenta αi to specify the block in (4.6). The

calculation is straightforward: The spins sa of the W algebra generators are

determined by the group theory [57]. They are given by the exponents of g

augmented by 1. For a spin s generator W (s)(z), the W-algebra can be used

to remove all but the s − 1 of its modes W(s)−s+m, where m runs from 1 to

s− 1. There is a global Ward identity on a genus zero Riemann surface that

imposes 2s − 1 linear relations between the insertions of W (s)’s at different

points, further reducing the multiplicity to k(s− 1)− (2s− 1) [57, 64]. For

ADE Lie algebras, the exponents are easily seen to satisfy

n∑a=1

(sa − 1) = h+, (4.13)

leading to the result we quoted.

4.2 Toda CFT, q-deformation and Wq,t(g)-algebra

TheW(g)-algebra can be defined as a complete set of currents that commute

with the screening charges. In [24], one constructed a deformation of both the

algebra and the conformal blocks in free field formulation, based on deforming

the screening currents. The screening current Sa(x) operators are deformed

so that

〈Sa(x), Sa(x′)〉free =

ϕq(ex−x′)

ϕq(t ex−x′)

ϕq(ex′−x)

ϕq(t ex′−x)

, (4.14)

and, for a 6= b

〈Sa(x), Sb(x′)〉free =

(ϕq(tvex−x′)ϕq(v ex−x

′)

)Iab(4.15)

where v = (q/t)12 . The explicit formulas the q-deformed screening charges S

are in [24], and we review them in the appendix A. The Wq,t(g) algebra is

defined in [24] as a set of all operators commuting with the deformed screening

charges (together with a set of screening charges with q and t exchanged).

37

After the q-deformation, the conformal block (4.7) becomes manifestly equal

to the partition function of the D3 brane gauge theory: the screening charge

contributions in (4.8), (4.9) to the conformal block are the contributions

of N = 4 vector and hypermultiplets in (3.7) and (3.8) to the D3 brane

partition function, where the number Na of D3 branes on a-th node maps to

the number of screening charge insertions.

The primary vertex operators in the q-deformed Wq,t(g) algebra are de-

formations of (4.4). A construction of q-deformed primary vertex operators,

with generic momenta, is given in the appendix A. They have the form

:n+1∏i=1

Vωi(xi) :→ Vα(z)

where ωi are a collection of n+1 weights of g, suitably chosen. See appendix

A and sectio 6 for explicit expressions. Each of the Vωi(x)’s has two point

functions with the screening operators that equal (3.9),

〈Sa(x)Vωi(f)〉free. (4.16)

Explicitly, Vωi(f) is a normal ordered product of fundamental vertex oper-

ators of the form W±1a (fvr) whose two point functions with the screening

charges are ϕ±1q (fvr/x), equal to the contributions to (3.9) of either chiral or

anti-chiral multiplets of R-charge r. To go back to the undeformed theory,

we can let q = exp(Rε1), t = exp(−Rε2), and take the R to zero limit. In the

limit, q and t tend to 1, where we take (2.25)

exi = zP qαi,P , (4.17)

(4.16) above becomes (1 − ex/zP)(ea,αP ) where αP =∑

wi∈WP xi wi. This

is the two-point function of the vertex operator with the screening charge

in Toda CFT, given in (4.11). In principle, one can consider insertions of

any collection WS of Vωi(x)’s with∑

ωi∈WS ωi = 0. The CFT limit of this

38

depends on the collectionWS , and how insertion scales in the R to zero limit:

one can get any collection of primary vertex operators with either arbitrary

or (partially) degenerate momenta.

After the q-deformation, theW(g) algebra is no longer a symmetry, since

W(g) is not a subalgebra ofWq,t(g), so the argument does not apply. Corre-

spondingly, instead of hypergeometric equations, the conformal blocks now

satisfy q-difference equations. The number of solutions of these can be larger,

as some linearly independent solutions to q-hypergeometric equations can be-

come linearly dependent in the q → 1 limit. We will see that this is indeed

what happens for g 6= An.

5 Triality

We will show that the partition function of (2, 0) little string on C with codi-

mension two defects equals the q-deformation of the Toda CFT conformal

block on C, with vertex operators determined by positions and types of de-

fects. Since we have shown, in section 6, that the q-deformed CFT correlator

(4.6) equals the 3d partition function in (3.6), we only need to show equality

of partition functions of the 3d gauge theory on D3 branes and the partition

function of the (2, 0) theory on C.The relation between the partition function of the A1 6d (2, 0) CFT and

the 2d Liouville CFT was conjectured by Alday, Gaiotto and Tachikawa in [5].

That conjecture was proven in [25, 26]. Generalization of the correspondence

for other groups were studied by many papers, see for example [38, 39, 67] and

[68] for a recent collection of reviews. For pure gauge theories of ADE type,

the relation between the gauge theory partition function and the norm of the

Whittaker vector of the W-algebra was proven recently in [69] (see [70] for

the q-deformed An version). An obstacle to extending the correspondence to

groups other than A1 is that compactification of (2, 0) 6d CFT on a Riemann

39

surface does not in general lead to a theory with a Lagrangian description:

without it, the partition function of the theory is not computable either, so

there is nothing to compare to the Toda conformal block. Another obstacle

is that, for g 6= A1, the general Toda conformal blocks are known only if they

admit free field representation; after q-deformation, this is true even in the

Virasoro case.

The little string perspective of the present paper is crucial to establish

the precise statement of the correspondence for arbitrary ADE groups and

general blocks on a sphere, and leads to a proof of it for conformal blocks ad-

mitting free field representation, generalizing [1] for An (and [27] for A1). The

correspondence between the (2, 0) theory and Toda theory in the conformal

limit follows by taking the ms to infinity limit.

5.1 Gauge/Vortex Duality

The relationship between the 5d N = 1 gauge theory on D5 branes in section

3, and the 3d N = 2 gauge theory on D3 branes in section 4 is gauge/vortex

duality. The duality comes from two different, yet equivalent descriptions of

vortices in the theory: one from the perspective of 5d theory with vortices,

and the other from the perspective of the 3d theory on the vortex. The

fact that the theory on vortices captures aspects of dynamics of the ”parent”

gauge theory was noticed early on in [71] at the level of BPS spectra. Turning

on Ω-background transverse to the vortex, the correspondence becomes more

extensive [72, 73]: it is a gauge/vortex duality [1, 27, 28].

The 2d Ω-background transverse to the vortex (with parameter ε2) is

necessary. It ensures that the super-Poincare symmetries preserved by the

5d and the 3d theories are the same, since the Ω-background is a form of

compactification [74–76] and breaks half the supersymmetry: after turning

it on, both theories are 3d N = 2 theories on a circle. The duality should

40

hold at the level of F-type terms – the Kahler potentials are not protected,

and we don’t claim to specify them. The duality is the little string analogue

[28] of large N dualities in topological string [29–32]. The D3 brane gauge

theory lives in the Higgs phase of the bulk theory. From the bulk perspective,

the theory starts out on the Higgs branch, but ends up pushed onto the

Coulomb branch due to the vortex flux. In the Higgs phase, the Coulomb

moduli are frozen to points where the hypermultiplets can get expectation

values. Turning on N units of vortex flux in a U(1) gauge group shifts the

corresponding Coulomb modulus a to a + Nε2, where ε2 is the parameter

of the Ω in background rotating the complex plane transverse to the vortex.

This is a consequence [1, 27, 28] of how Ω-background deforms the Lagrangian

of the 5d theory [18, 77]. Once we have a pair of dual theories, one expects

that their partition functions in the full Ω-background, depending on ε1,2,

agree as well. This was shown in detail in [1, 27] for the A1 and the An

theories. The generalization to the ADE case works in an analogous way, so

we will be brief.

5.2 Equality of Partition Functions

In the partition function of D3 branes in (3.5), we choose an integration cycle,

corresponding to choosing a vacuum of the theory. Evaluating the contour

integral, we pick up all the poles from (3.6) within it. The poles turn out

to be labeled by touples of 2d partitions R, with one row per integration

variable. This allows us to express the contour integral in (3.5) as a sum over

2d partitions

Z3d =

∮dx I3d(x) =

∑R

resRI3d(x). (5.1)

The residue at the pole labeled by R, normalized by the residue of the pole

at ∅, corresponding to trivial 2d partition, is simply equal to the ratio

resR I3d(x)/res∅I3d(x) = I3d(xR)/I3d(x∅) (5.2)

41

of the integrands at the two points, which is finite and simple to compute.

From the bulk perspective, the choice of integration cycle in (3.5) corre-

sponds to picking the values of the Coulomb moduli where (2.14) gets eval-

uated, or more precisely, choosing the vortex fluxes which determine them.

Evaluating the 5d partition function at these values (5.6), one finds that it

simplifies: only the partitions with finite numbers of rows, determined by the

fluxes, contribute. The equality of the bulk partition function in (2.14)

Z5d = r5d

∑R

I5d,R(e) (5.3)

and the D3 brane partition function follows by noticing that term by term,

contributions of a tuple R to (2.14) equals to the residue in (5.1):

I5d,R = I3d(xR)/I3d(x∅). (5.4)

From the Toda CFT point of view, this corresponds to specifying the confor-

mal block; for g 6= A1, there is a finite dimensional space of conformal blocks

to choose from, even for the 3-point function, see section 4.

Bulk Partition Function

The bulk partition function (2.14) is written in terms of the basic building

block, the Nekrasov function in (2.15). We can rewrite the function as

NR1R2

(e1

e2

)=

N1∏i=1

N2∏j=1

ϕq(e1e2qR1,i−R2,j tj−i+1

)ϕq(e1e2qR1,i−R2,j tj−i

) ϕq(e1e2tj−i)

ϕq(e1e2tj−i+1

)NR1,∅

(tN2

e1

e2

)N∅,R2

(t−N1

e1

e2

)assuming that the partitions R1 and R2 have no more that N1 and N2 rows,

respectively. Let’s suppose that a partition R(a),I in (2.14) has no more

than N(a),I rows. Then, using the property just quoted, we can re-write the

42

contributions of vector and hypermultiplets to (2.14) as follows:

n∏a=1

zVa, ~R(a)=

n∏a=1

I3da (x~R(a)

)

I3da

(x~∅(a)

) · Vvect,

where I3d,Va is the contribution of the 3d vector multiplet corresponding to

a U(N(a)) gauge group in (3.7) evaluated at positions

ex~R(a) = e(a),I q

R(a),I tρv#a, (5.5)

determined by the lengths of the rows of the partitions Ra,IdaI=1. Here, v#a is

a power of v that depends on the position of the node in the Dynkin diagram:

it is equal to 1 for a = 1, and increases by one with every link. We labeled

the remaining factor Vvect. Similarly, the contributions of 5d bifundamentals

can be written as

n∏a,b=1

zHa,b, ~R(a), ~R(b) =n∏

a,b=1

[I3dab

(x~R(a)

, x~R(b)

)I3dab

(x~∅(a)

, x~∅(b)

)]Iab · Vbifund

where I3dab

(x(a), x(b)

)Ia,b is the contribution of the 3d bifundamental hypermul-

tiplet multiplet, and Vbifund stands for all the remaining factors. We write

the contributions of fundamentals asn∏a=1

zHa, ~R(a) = Vfund

and∏n

a=1 zCS,~R(a) = VCS, for the 5d Chern-Simons contribution. This lets

us write (2.14), the contribution of a the couple R to the bulk partition

function, as

I5d,R =n∏a=1

I3da (x~R(a)

)

I3da

(x~∅(a)

) · n∏a,b=1

[I3dab

(x~R(a)

, x~R(b)

)I3dab

(x~∅(a)

, x~∅(b)

)]Iab · VvectVbifundVfundVCS

This is merely a rewriting of (2.14), in particular, the sum runs over arbitrary

tuples R. However, several remarkable things happen if we set e(a),I equal

to fi tN(a),I , with a suitable proportionality constant:

e(a),I = fi tN(a),I v#a,i,I . (5.6)

43

Firstly, the product VvectVbifundVfundVCS vanishes if partition R(a),I in the tuple

has more than N(a),I rows. Secondly, there are many cancellations in the

product, which simplifies to

VvectVbifundVfundVCS =n∏a=1

[ I3da,V

(x~R(a)

)I3da,V .(x~∅(a)

)] (5.7)

The summands of the bulk partition function (2.14) turn into the D3 brane

partition function,

I5d,R =n∏a=1

I3da (x~R(a)

)

I3da

(x~∅(a)

) · n∏a,b=1

[I3dab

(x~R(a)

, x~R(b)

)I3dab

(x~∅(a)

, x~∅(b)

)]Iab · n∏a=1

[I3da,V

(x~R(a)

)I3da,V

(x~∅(a)

)] (5.8)

evaluated at discrete set of points x = x~R(a), determined from (5.6) and

(5.5). The right hand side is the contribution of poles in evaluating the 3d

partition function by contours. The family of points x = x~R(a), indexed by

2d partitions of finite length, are the positions of the poles in the integral.

This leads to the third remarkable fact: The subtle v factors are related to

the SR R-symmetry charges of 3d chiral multiplets. These should be fixed by

requiring superconformal invariance of the 3d theory in the IR. At the same

time, they turn out to be determined by the Wq,t(g) algebra symmetry in

[24]: the choice of the factors that equates the residues with the 5d partition

function is exactly the same one as what is needed for the partition function

to equal the q-deformed conformal block of the Toda CFT. The powers v#a,i,I

in (5.6) can be read of directly from the vertex operators.

3d Partition Function

The contours of integration in (5.1) are associated to vacua of the 3d

theory in flat space. The poles in (5.5) that contribute to the contours

correspond to supersymmetric vacua of the 3d gauge theory in Ω-background

along the D3 brane, depending on q.

44

Rather than giving detailed examples of contours (which one can find in

[1, 27]), let us describe the geometry behind them. In a vacuum of the 3d

gauge theory, bifundamentals and chiral matter fields get expectation values,

due to FI terms which are turned on. This has a geometric interpretation

which is very useful [32]. Giving expectation values to bifundamentals cor-

responds to binding the D3 branes into 2-cycles whose homology classes in

H2(X,Z) = Λ are positive roots eγ. The restriction to positive roots comes

from supersymmetry, which requires us to take positive integer combinations

of D3 branes eγ =∑

a γa ea, with γa ≥ 0. In the vacuum with FI terms,

chiral multiplets from D3-D5 strings must also have expectation values; this

corresponds to ending the D3 branes on a D5 brane which is wrapping a

cycle S∗i in WS . The 2-cycles Dγ,i,α one gets in this way have homology

classes corresponding to positive root vectors eγ and have a boundary on a

D5 brane wrapping ωi = [S∗i ]. Depending on a sign of the Fayet Iliopolous

term, only the chiral multiplets in fundamental representation can get ex-

pectation values. The supersymmetric vacuum describes distributing the D3

branes between the cycles Dγ,i,α, satisfying the conditions that all the wrap-

ping numbers are positive, and that D3 brane charge is conserved. (The

index α allows for the possibility that there can be more than one such cycle.

We will see such examples in section 6.) A chiral multiplet that gets ex-

pectation value is reflected in poles at points on the Coulomb branch where

it becomes massless. In the Ω-background along the D3 branes, one gets

families of such vacua shifted by the values of the 3d vortex fluxes along the

D3 branes, so the vacua come in families, parametrized by the lengths of

the rows of 2d partitions, one for each 3d Coulomb modulus x. For the An

and Dn examples, we have checked explicitly that there is indeed a choice of

contours reflecting this structure, and such that evaluating the integral by

residues leads to (5.1), with poles at (5.5),(5.6).

45

6 Examples

We will now illustrate the general results of previous sections with the basic

case of the little string theory on a sphere with three full punctures, in the

terminology of [3]. This corresponds to a (q-deformed) 3-point function for

Toda CFT, with 3 primary operator insertions of generic momenta. In little

string theory on a Riemann surface C which is a cylinder, there are two

punctures to begin with. The third puncture is introduced by D5 brane

defects.

To construct the full puncture defect, as explained in section 2.4, one picks

a collection of n + 1 weight vectors WS = ωin+1i=1 satisfying the following

properties: ωi’s are chosen from the Weyl orbits of the n fundamental weights

of g (or more precisely, of minus the fundamental weights), they provide a

basis of the weight lattice Λ∗, and sum up to zero

n+1∑i=1

ωi = 0. (6.1)

From the data given we can find the three theories related by triality, as

we explained in section 5: the 5d gauge theory, the D3 brane gauge theory

and the collection of vertex operators in the q-deformed Toda CFT they

correspond to.

The parameters of the theory are as follows: for each ωi, we pick a point

on C with coordinate xi = Rmi. xi encodes the position of the D5 brane

wrapping ωi = [S∗i ] on C, and the masses mi’s of matter fields in the 5d and

the 3d gauge theory. From Toda perspective, they correspond to n momenta

and the the position of the puncture. With only 3 punctures, the latter can

be set to z = 1. In addition we need to specify n moduli of (2, 0) theory

we called τa’s in (2.10). They determine the 5d gauge couplings, the 3d

FI parameters and correspond to the momentum of the puncture at z = 0

from the Toda perspective. There are n more parameters to specify: the n

46

non-normalizable Coulomb moduli associated with the U(1) centers of the

5d gauge groups in (2.11); the net ranks Na of the 3d gauge groups in (3.2),

and the numbers of screening charges in (4.3) in Toda theory, or equivalently,

the momentum of the puncture at z =∞.

5d Gauge Theory

We split each of the weights ωi into −wa plus a sum of simple roots with

non-negative coefficients. Here, wa is the fundamental weight in whose orbit

−ωi lies. This splitting is unique, due to a well known theorem in the theory

of Lie algebras and their representations.9 This corresponds to splitting of

the cycles wrapped by non-compact D5 branes on the Higgs branch, into a

canonical non-compact part and the remaining collection of compact cycles.

We collect all the non-compact cycles into [S∗] = −∑n

a=1 mawa, where ma

counts how many ω’s came from the orbit of −wa, and the compact cycles

into [S] =∑n

a=1 daea. As a consequence of (6.1), these satisfy the constraint∑bCabdb = ma. This leads to a 5d N = 1 ADE quiver gauge theory with

U(da) gauge group and ma fundamental hypermultiplets on the a’th node, to-

gether with the bifundamental matter fields making up the Dynkin diagram

of the Lie algebra. From the quiver, we can compute the partition func-

tion bulk theory as the Nekrasov partition function of the 5d gauge theory,

following the prescription in section 2.2.

3d Gauge Theory

We take D3 branes in class [D] =∑n

a=1Naea. The 3-3 strings lead to

the N = 4 quiver theory with gauge group∏n

a=1 U(Na) and bifundamental

hypemultiplets according to the Dynkin diagram of g. The N = 2 matter

comes from 3-5 strings: strings stretching between the D5 brane on S∗i and

9We thank Chrisitan Schmidt for collaboration relating to this point.

47

the D3 branes on Sa give (ωi, ea) anti-chiral minus the chiral multiplets in

fundamental representation of the U(Na) gauge group. From the quiver,

we can compute the partition function of the 3d gauge theory, following

the prescription in section 3.3. The partition function depends also on the

relative splitting of the Na’s between the vacua. For each positive root eγ,

we get as many vacua as ωi’s with (eγ, ωi) < 0, counted with multiplicity

|(eγ, ωi)|. This number, as we’ll see, always equals the number of Coulomb

moduli of the 5d gauge theory.

Toda CFT

The collection of weights ωi satisfying (6.1) leads to the q-deformed pri-

mary vertex operator Vα1(z)

:n+1∏i=1

Vωi(xi) : → Vα1(z) (6.2)

where xi, z, and α, are related as in (4.17), exi = z qαi . With only 3

punctures, no physical quantity will depend on z itself, so we can set it to 1.

The vertex operator Vωi(x) corresponding to the weight ωi is constructed in

appendix A. This vertex operator is the q-deformation of the primary Vα1(z).

Taking the vertex operators to z = 0,∞, the details of the q-deformation

are not important since only the zero modes survive and these are already

encoded in τa and Na. The general form of the vertex operator is given in

the appendix; specific operators for all the ADE groups will be given below.

The vertex operators Vωi(xi) encode the R-charges of the 3d chiral multiplets

coming from strings with one end-point on the D5 brane wrapping S∗i , and

we can read off from them the subtle v factors in (5.6).

In Toda CFT, the 3-point function of W-algebra primaries

〈Vα0(0)Vα1(1)Vα∞(∞)〉 (6.3)

48

is labeled by the three momenta α0, α1, α∞. If α∞ = −α0−α1−∑n

a=1Naea/b

for positive integers Na (the ranks of D3 brane gauge theory) we can compute

the three-point function (6.3) in free field formalism where we insert Na

screening charge generators

〈Vα0(0)Vα1(1)Vα∞(∞)n∏a=1

QNaa 〉free. (6.4)

Replacing the vertex operators and the screening charges Qa =∫dxSa(x)

by the q-deformed ones, we get the q-deformed 3-point conformal block of

Wq,t(g) algebra, as described in section 4 and appendix A. The q-deformed

block is the partition function of the gauge theory on D3 branes given by the

corresponding data. In particular, the number of q-deformed blocks of the

Wq,t(g) algebra is the number of branches of the 3d gauge theory. Computing

(6.4) by residues gives the bulk partition function, as we explained in sec. 5.

Below, we will make this explicit for all the ADE groups, beginning with

the familiar g = An case, where we will recover the results obtained in [1]

using a different, T -dual setting. We will move on to g = Dn, En theories,

where the technique of the present paper are indispensable. Before we do

that, let us digress and discuss the CFT limit of the theory.

Coulomb Moduli, Contours and Fusion Multiplicities

Since the description of defects in little string is new, and the q-deformed

Toda CFT’s are not familiar to many, we will pause to demonstrate that

in the ms to infinity limit the 5d gauge theory describes the (2, 0) CFT on

the 3-punctured sphere, and show how the correspondence to the Toda CFT

3-point conformal block emerges. The D5 branes source the Higgs field of

the (2, 0) little string theory which, on the Higgs branch, takes the form (6.5)

eR′ϕS(x) = eτ

∏ω∨i ∈WS

Vi(x)ω∨i . (6.5)

49

We defined10

Vi(x) = (1− ex−miR′)−1,

see sections 2.2.2-3. In the ms to infinity limit we described in 2.4.1, the

above becomes

ϕS(z) =α0

z+

α1

1− z, (6.6)

where

α1 =n+1∑i=1

mi ω∨i , α0 = τ/R′,

and z = e−x. This makes it manifest we have (2, 0) theory with three punc-

tures on C. The punctures at z = 0,∞ are there because C is a cylinder and

ϕS = ϕS(z)dz a one form on it, with values in the Lie algebra; the puncture

at z = 1 comes from D5 branes. The residues α0 and α1 are both generic

elements of the Cartan subalgebra of g. For α1, this is the case since the n+1

weights in WS satisfy a single algebraic relation (they sum up to zero) and

the D5 brane positions are taken to be generic as well. For α0, this is true

because we needed all n gauge couplings τa to be arbitrary complex numbers

with Re(τa) > 0. From the bulk perspective, the momentum at infinity α∞ is

determined by the n non-normalizable Coulomb moduli associated with the

n U(1) centers of the U(da) gauge groups: the U(1) gauge fields are frozen

by the Green-Schwarz mechanism, and the Coulomb moduli that come with

them are parameters, not moduli.

As we explained in section 4, to fully specify the conformal block, we need

to specify h+(g)−n more parameters coming from fusion multiplicities. This

matches the number of Coulomb moduli of the bulk theory: we explained in

section 2.4.1 that one finds h+(g)−n normalizable Coulomb. This is also the

same as the number of types contours of integration of the D3 brane theory

in the limit, or equivalently, the number of choices of vacua of the theory. In

10The exponent in ω∨i is to remind us to interpret them as linear combinations of the

Chevalley generators of the Cartan subalgebra, see section 2.

50

[32, 49], the number of vacua of an ADE quiver gauge theory with superpo-

tential TrWa(Xa) for the adjoint chiral multiplet, was studied. It was shown

that, if Wa(x) has an isolated critical point for each a, there are h+(g)−n of

vacua corresponding to different breaking patterns of the∏n

a=1 U(Na) gauge

group. In a vacuum, the gauge group is broken to∏

γ>0 U(Nγ) where eγ

are positive roots. One has h+(g) integers to specify, the number of positive

roots, but there are n constraints since this has to come from the origi-

nal gauge group:∑

γ>0Nγ eγ =∑n

a=1 Na ea. This problem relates to ours

by thinking of eigenvalues of Xa as parameterizing the positions of the D3

branes on C, and integrating out the chiral N = 2 matter to get TrWa(Xa)

as the effective superpotential. In our context, is easy to show that the ef-

fective superpotential satisfies ∂xWa(x) = (ea, ϕS(x)) [28, 74, 75], and has

one critical point for each a. To establish more detailed correspondence with

the D3 brane theory or to Toda CFT, one has to go back to the little string,

since no other way is known to compute the partition function of the (2, 0)

CFT on C.After the q-deformation, the number of Coulomb branch moduli of the

bulk theory, the vacua of the 3d gauge theory on D3 branes, and the contours

of integration of q-deformed conformal blocks are equal to each other in all

ADE theories, but they are larger than h+(g)−n, except for g = An. As we

explained earlier, this is not surprising from neither the bulk nor the Toda

perspective, as the massive theory contains more data than its CFT limit.

In 5d gauge theory, the number of normalizable Coulomb moduli is simply∑na=1(da−1). From the 3d gauge theory perspective, counting vacua is more

complicated, and it is remarkable that one always finds agreement. The

vacua are still related to counting positive roots, but now with multiplicity,

as we explained in section 5.2. In the vacuum, the D3 brane gauge group is

broken to∏

Dγ,i,αU(Nγ,i,α) where we get a gauge group factor for each cycle

51

[Dγ,i,α] = eγ with boundary on [S∗i ] = ωi. We get |(eγ, ωi)| = Iγ,i of such

cycles if (ea, ωi) is negative – we used the index α to label them. If (ea, ωi)

is not negative, we get none. Thus, to specify the breaking pattern, we get

to pick an integer for each of the positive roots eγ, counted with multiplicity∑ωi∈WS Iγ,i. The integers have to satisfy n constraints for the total ranks

to add up to Na, for each a. This count also equals the number of choices

in specifying the contour of integration in (3.5), and the number of fusion

multiplicities ofWq,t(g) algebra, since the same integral computes both (3.5)

and (6.4), after q-deformation.

6.1 An Little String Theory

For the n+ 1 weight vectors ωi = [S∗i ] we choose:

ω1 = −w1,

ω2 = −w1 + e1,...

ωn = −w1 + e1 + . . .+ en−1,

ωn+1 = −w1 + e1 + . . .+ en−1 + en.

(6.7)

Here, w1 is the highest weight of the 1st-fundamental representation; the

rest of the weights on the right hand side of (6.7) are obtained from it by

acting with the Weyl group of An; the weights −ωi are the n + 1 weights

of the defining, n + 1 dimensional representation of An (”the fundamental

representation”). An equivalent way to write (6.7) is in terms of fundamental

weights, using ea =∑n

b=1Cabwb:

ω1 = −w1,

ωi = −wi + wi−1, i = 2, . . . n,

ωn+1 = wn

(6.8)

The set of weights WS = ωin+1i=1 spans the weight lattice and satisfies (6.1).

52

6.1.1 The 5d An Gauge Theory

To find 5d gauge theory description of the (2, 0) little string, we sum up the

simple roots on the right and side of (6.7) to get [S], the homology class of

D5 branes wrapping compact 2-cycles:

[S] = ne1 + (n− 1)e2 + . . .+ en

This leads to an An quiver theory where the coefficient of ea in [S] is the

rank of the gauge group on the a-th node, da = n− a+ 1. The fundamental

matter comes from non-compact D5 branes whose homology class [S∗] is the

sum of (minus) the fundamental weights in (6.7)

[S∗] = −(n+ 1)w1.

To number of hypermultiplets transforming in fundamental representation of

the gauge group on the a-th node is the coefficient of −wa in the expansion of

[S∗] in terms of fundamental weights. In this case, we have n+ 1 hypermul-

tiplets in fundamental representation of the gauge group on the first node.

Altogether, we get the 5d N = 1 quiver gauge theory in Fig. 2. This agrees

with the TN theory shown in [1] to describe the 3-point function of An with

N = n+ 1, using a dual string realization. Given the lagrangian description

of the low energy theory, the partition function can be obtained using the

prescription in section 2.2. The 5d TN theory was also obtained in [13, 14].

6.1.2 The 3d An Gauge Theory

We start with the collection of D5 branes wrapping n+1 non-compact cycles

S∗i with [S∗i ] = ωi, arising in the Higgs phase of the 5d gauge theory. We add

the D3 branes wrapping the compact two-cycles in the homology class of D:

[D] =n∑a=1

Na ea.

53

Figure 8: D3 quiver from An little string.

The strings beginning and ending on D3 branes lead to the N = 4 An quiver

theory with gauge group∏n

a=1 U(Na) and bifundamental hypermultiplets

according to the An Dynkin diagram. The N = 2 matter comes from inter-

sections of D3 branes with D5 branes in WS . The number of chiral minus

the anti-chiral multiplets in fundamental representation of the gauge group

on the a-th node, coming from a D5 brane on ωi, is

#(Sa, S∗i ) = (ea, ωi) = −δa,i + δa,i−1,

as we explained in section 3.1. This is the coefficient of wa in the expansion

of ωi in terms of the fundamental weights in (6.10). The theory on the D3

branes is an An N = 2 quiver theory in Fig. 8. This agrees with the quiver

of vortex theory derived in [1] using different means. From the quiver, we

can compute the partition function of the 3d gauge theory, following the

prescription in section 3.2.

6.1.3 An Toda Conformal Block

The q-deformed vertex operator is :∏n+1

i=1 Vωi(xi) : where

Vω1(x) = W−11 (x),

Vω2(x) =: W−11 (x)E1(xv−1) :

...

Vωn(x) =: W−11 (x)E1(xv−1)E2(xv−2) . . . En−1(xv1−n) :

Vωn+1(x) =: W−11 (x)E1(xv−1)E2(xv−2) . . . En−1(xv1−n)En(xv−n) :

(6.9)

54

in terms fundamental weight and simple root vertex operators in the ap-

pendix A. This way of writing the vertex operators encodes the subtle v

dependence of the Coulomb moduli at the triality point in (5.6). The vertex

operators can be thought of as quantizing the classical formula (6.7). We

have e(a),I = fi tN(a),I v−2a+1, where I runs from 1 to n − a + 1. This comes

with a factor v−a × v−a+1 = v−2a+1, independently of which ωi the simple

root ea gets assigned to. The factor v−a comes from (6.9) and is the subtle

one; the second factor v#a = v−a+1 can be read off from the Dynkin diagram

as defined in section 5.2. This reflects the Wq,t(An) algebra symmetry of the

theory in Ω background.

There is an equivalent way of writing the vertex operators (6.9):

Vω1(x) = W−11 (x),

Vωi(x) = W−1i (xv−i+1)Wi−1(xv−i), i = 2, . . . n,

Vωn+1(x) = Wn(xv−n−1).

(6.10)

Written this way, the vertex operators encodes the R-charges of chiral mul-

tiplets. The U(1)R-charge SR = −r/2 of an (anti)chiral multiplet from 5-3

strings ending on D5 brane in class ωi and the D3 brane on the a’th node

is encoded in the v-shift of the argument of Wa(xvr). The vertex opera-

tors, essentially in this form, appeared in [24] in section 5.1.1, with obvious

translations.

The number of normalizable Coulomb moduli of the 5d theory is∑n

a=1(da−1) = h+(An) − n, so the dimension of the Coulomb branch is the same in

the little string and in the (2, 0) CFT. This also matches the number of

vacua/contours of integration in (3.5), which can be counted using the pre-

scription in section 5.2: The positive roots eγ of the An Lie algebra are of

the form ei + ei+1 + . . . + ej, for i < j. Each positive root contributes with

multiplicity 1, since each root has negative multiplicity with exactly of the

weights in WS , and the intersection equals to −1. For the root we wrote,

55

the corresponding weight is ωi. Thus the breaking pattern of∏n

a=1 U(Na)

allows for h+(An)−n choices. This is the same as the data needed to specify

the q-deformed Wq,t(An) algebra 3-point block. As we explained in 4 this

number could have been greater, in principle, than the number of 3-point

functions of the An Toda CFT. In this case, it is not: The W(An) algebra

symmetry is generated by currents of spins sa = 2, 3, . . . , n + 1 leading to∑na=1(sa − 2) = 1

2n(n+ 1) − n = h+(An) − n additional parameters needed

to specify the conformal block.

6.2 Dn with 3 Full Punctures

For WS , we take the following collection of n+ 1 weights of Dn

ω1 = −w1 + e1 + e2 + . . .+ en−2 + en−1 + en,

ωi = ωi−1 + en−i, i = 2, . . . n− 1

ωn = −wn−1

ωn+1 = −wn

(6.11)

where wa is the a’th fundamental weight. The weights w1, wn−1, and wn, are

respectively the highest weights of the 2n dimensional defining representation

of Dn and the two 2n−1 dimensional spinor representations. The first set of

n−1 weights in (6.11) are all in the orbit of w1. Another way to write (6.11)

is as follows:

ω1 = −wn−2 + wn−1 + wn,

ωi = −wn−i−1 + wn−i, i = 2, . . . n− 1,

ωn = −wn−1,

ωn+1 = −wn

(6.12)

This way of writing ω’s makes it easy to check that the set WS spans the

weight lattice, and that ωi’s sum up to zero.

56

6.2.1 The 5d Dn Gauge Theory

To find the 5d gauge theory description of the (2, 0) little string, we sum up

the simple roots on the right and side of (6.11):

[S] = ne1 + (n+ 1)e2 + . . .+ (2n− 3)en−2 + (n− 1)en−1 + (n− 1)en.

The rank of the a’th gauge group in the Dn quiver diagram is the coefficient

of ea. The fundamental matter comes from non-compact D5 branes whose

homology class [S∗] is the sum of (minus) the fundamental weights in (6.11)

[S∗] = −(n− 1)w1 − wn−1 − wn.

The number of fundamental hypermultiplets on the a-th node is computed by

the coefficient of −wa in [S∗]. We get n− 1 hypermultiplets in fundamental

representation of the gauge group on the first node, and one each for the

n − 1’st and the n’th node. The resulting 5d gauge theory has a quiver

diagram given in figure Fig.3. It satisfies the condition∑

bCab db = ma. From

the quiver, the partition function of the (2, 0) theory follows immediately,

applying the formalism of section 2.2.

6.2.2 The 3d Dn Gauge Theory

We take the D3 branes to wrap the collection of compact two-cycles in the

homology class of [D] =∑n

a=1Na ea. The N = 2 matter comes from inter-

sections of D3 branes in class ea with D5 branes in class ωi; the number

of anti-chiral multiplets minus the chiral ones is the coefficient of wa in the

expansion of ωi in (6.12). This leads to the 3d quiver in figure 9. From

the quiver, we can compute the partition function of the 3d gauge theory,

following the prescription in section 3.2.

57

Figure 9: D3 quiver from Dn little string.

6.2.3 Dn Toda conformal block

Corresponding to weights in (6.11) we get the vertex operator :∏n+1

i=1 Vωi(xi) :,

with

Vω1(x) =: W−11 (x)E1(xv−1)E2(xv−2) . . . En−2(xv−n+2)En−1(xv−n+1)En(xv−n+1) :,

Vωi(x) =: Vωi−1(x)En−i(xv

−n−i+2) :, i = 2, . . . n− 1

Vωn(x) =: W−1n−1(x) :

Vωn+1(x) =: W−1n (x), :

(6.13)

in terms fundamental weight and simple root vertex operators in the ap-

pendix A. This way of writing the vertex operators encodes the subtle v

dependence of the Coulomb moduli at the triality point in (5.6). The vertex

operators can be thought of as quantizing the classical formula (6.7). Each

Coulomb modulus e(a),I gets assigned to one and only one simple root vertex

operator Ea in (6.14). This map also encodes the ωi to which the simple root

ea gets assigned to. We can read off that e(a),I = fi tNa,Iv#a,i,I , where v#a,i,I

is equal to v#′a,i,I , which enters the argument of the vertex operator Ea, times

v#a defined in section 5.2. The power of v in v#a counts the position of the

a-th node in the Dynkin diagram, in terms of number of links that separate

it from the first node, so that v#a = v1−a, for a between 1 and n − 2, and

v#n−1 = v#n = v2−n.

58

The vertex operators can also be written in terms of fundamental weight

vertex operators alone, using results from the appendix A. This gives

Vω1(x) =: Wn−2(xv−n+1)−1Wn−1(xv−n)Wn(xv−n) :

Vωi(x) =: W−1n−i−1(xv−n−i+2)Wn−i(xv

−n−i+1) :, i = 2, . . . n− 1

Vωn(x) =: W−1n−1(x) :

Vωn+1(x) =: W−1n (x) :

(6.14)

This way of writing the vertex operators encodes the U(1)R symmetry charges

of chiral multiplets, in the same manner as in the An case. This should cor-

respond to the 3d theory being conformal in the IR, although we have not

attempted to check that. The vertex operators Vωi , coming from the funda-

mental representation of Dn, with highest weight w1, are written, essentially

in this form, in [24] in section 5.1.4 of the paper.

The Dn Toda CFT has W(Dn) algebra symmetry generated by cur-

rents of spins sa = 2, 4, . . . , 2(n − 1), n. To fully specify the 3-point con-

formal block of the Toda CFT in (6.4), one has to specify the additional∑na=1(sa − 1) = n2 − n = h+(Dn) parameters, besides α0 and α1. This is

also the number of Coulomb moduli of the bulk theory and the D3 brane

vacua in the limit where ms goes to infinity. In the little string case, all

these numbers are larger: The number of Coulomb moduli of the bulk the-

ory is∑n

a=1(da − 1) =1

2(n − 1)(3n − 2) − n. This is also the number of

integers we get to pick to specify the vacuum of the 3d theory, obtained

by counting positive roots, with multiplicity:(n− 1)(n+ 2)

2positive roots

have come with multiplicity 1 as they have intersection number −1 with one

of the weights, and(n− 1)(n− 2)

2positive roots count with multiplicity 2,

as they have intersection number −1 with two of the weights. Fortunately,(n− 1)(n+ 2)

2+ 2 · (n− 1)(n− 2)

2=

1

2(n− 1)(3n− 2), leading to the same

count as the Coulomb moduli.

59

6.3 E6 with 3 Full Punctures

For the weight system WS , we take:

ω1 = −w5

ω2 = −w5 + e5

ω3 = −w5 + e1 + 2e2 + 3e3 + 3e4 + 2e5 + 2e6

ω4 = −w5 + e1 + 2e2 + 4e3 + 3e4 + 2e5 + 2e6

ω5 = −w5 + e1 + 3e2 + 4e3 + 3e4 + 2e5 + 2e6

ω6 = −w5 + 2e1 + 3e2 + 4e3 + 3e4 + 2e5 + 2e6

ω7 = −w6

(6.15)

The first 6 weight vectors in (6.15) can be seen to lie in the Weyl orbit of

−w5. This can be rewritten as

ω1 = −w5

ω2 = −w4 + w5

ω3 = −w3 + w4 + w6

ω4 = −w2 + w3

ω5 = −w1 + w2

ω6 = w1

ω7 = −w6

(6.16)

which makes it easy to check that WS provides a basis of the weight lattice

of E6 and the weights in WS sum up to zero.

6.3.1 The Bulk E6 Gauge theory

Summing up the fundamental weight and the simple root part of the right

hand side in (6.15) separately, we find

[S] = 5e1 + 10e2 + 15e3 + 12e4 + 9e5 + 8e6. (6.17)

and

[S∗] = −6w5 − w6 (6.18)

60

Figure 10: D3 quiver from E6 little string.

This leads to a 5d quiver gauge theory description in figure 4. The partition

function of the theory can be computed from the quiver, as in section 2.2.

6.3.2 The D3 Brane E6 Gauge theory

The N = 2 matter can be read off from (6.16). This leads to the 3d quiver

in figure 10.

6.3.3 E6 Toda CFT

The E6 vertex operators corresponding to (6.15) are the following:

Vω1(x) = : W−15 (x) :,

Vω2(x) = : W−15 (x)E5(xv−1) :,

Vω3(x) = : W−15 (x)E5(xv−1)E4(xv−2)E3(xv−3)E2(xv−4)E6(xv−4)E1(xv−5)

E3(xv−5)E2(xv−6)E4(xv−6)E3(xv−7)E5(xv−7)E4(xv−8)E6(xv−8) :,

Vω4(x) = : Vω3(x)E3(xv−9) :,

Vω5(x) = : Vω4(x)E2(xv−10) :,

Vω6(x) = : Vω5(x)E1(xv−11) :,

Vω7(x) = : W−16 (x) :

(6.19)

From this, we can read off the Coulomb moduli in (5.6): we map e(a),I in one

to one way to the Ea vertex operators in (6.19), and set e(a),I = fitN(a),Iv#a,i,I ,

61

where fi corresponds to the ωi to which we assign this simple root. The v-

factor is encoded in the vertex operator (6.19), we just need to shift the

power of v in the argument of the vertex operator by v#a .

Rewriting (6.19) in terms of the fundamental vertex operators, using the

relations (A.1),

Vω1(x) =: W−15 (x) :,

Vω2(x) =: W5(xv−2)W−14 (xv−1) :,

Vω3(x) =: W4(xv−9)W6(xv−9)W−13 (xv−8) :

Vω4(x) =: W3(xv−10)W−12 (xv−9) :,

Vω5(x) =: W2(xv−11)W−11 (xv−10) :,

Vω6(x) =: W1(xv−12) :,

Vω7(x) =: W−16 (x) :

(6.20)

These encode the U(1)R-charges of chiral multiplets. The (anti-)chiral multi-

plet in fundamental representation of the a’th gauge group and coming from

the D5 brane on ωi has R-charge SR = −r/2, where vr is the v-dependence

in the argument of the vertex operator W±1a (xvr) on the right hand side of

Vωi(x) in (6.20).

The number of positive roots of E6 is h+(E6) = 36. The spins of the

generators of W(E6) algebra are sa = 2, 5, 6, 8, 9, 12. In the W(E6) algebra,

to specify the 3-point block, we need∑6

a=1(sa − 2) = 36 − 6 parameters

besides α0,1,∞. The number of Coulomb moduli of the 5d theory, including

the non-dynamical U(1) factors, is 59−6. This is also the number of integers

we need to specify the vacuum of the 3d theory: the latter is obtained by

counting positive roots with multiplicity. The 36 positive roots of E6 split

up into 19 positive roots with multiplicity 1; 13 positive roots with multi-

plicity two due to having intersection number −1 with two of the weights; 1

positive root with multiplicity 2 due to intersection number −2 with one of

the weights, 4 positive roots have intersection number −1 with three of the

62

weights. This gives 59 altogether, and we still have to subtract 6 for the net

rank constraints.

6.4 E7 with 3 Full Punctures

For the weight system WS , we take:

ω1 = −w1 + 3e1 + 5e2 + 7e3 + 6e4 + 4e5 + 2e6 + 4e7

ω2 = −w1 + 3e1 + 5e2 + 8e3 + 6e4 + 4e5 + 2e6 + 4e7

ω3 = −w1 + 3e1 + 6e2 + 8e3 + 6e4 + 4e5 + 2e6 + 4e7

ω4 = −w1 + 4e1 + 6e2 + 8e3 + 6e4 + 4e5 + 2e6 + 4e7

ω5 = −w6

ω6 = −w6 + e6

ω7 = −w6 + e5 + e6

ω8 = −w7

(6.21)

The first four weights are in the Weyl orbit of −w1, the next three in the

orbit of −w6. We can rewrite this as

ω1 = −w3 + w4 + w7

ω2 = −w2 + w3

ω3 = −w1 + w2

ω4 = w1

ω5 = −w6

ω6 = −w5 + w6

ω7 = −w4 + w5

ω8 = −w7

(6.22)

from which it is easy to check thatWS spans the weight lattice and that ωi’s

sum up to zero.

63

6.4.1 The Bulk E7 Gauge Theory

To find the 5d gauge group, we sum up the simple roots on the right hand

side of (6.21) to get:

[S] = 13e1 + 22e2 + 31e3 + 24e4 + 17e5 + 10e6 + 16e7, (6.23)

The fundamental weights sum up to

[S∗] = −4w1 − 3w6 − w7, (6.24)

This leads to the quiver gauge theory in figure 5, from which we can compute

the bulk partition function, using methods of section 2.2.

6.4.2 The 3d E7 Gauge Theory

To read off the N = 2 chiral matter content of the 3d quiver gauge theory

on the D3 branes, we use the second way of writing the weight system. The

Figure 11: D3 quiver from E7 little string.

number of N = 2 (anti)-chiral multiplets in the fundamental representation

of the gauge group on the a-th node, coming from D5 branes in class ωi,

is the coefficient of −wa in the expansion of ωi in (6.22). This leads to the

quiver in figure 11.

64

6.4.3 E7 Toda CFT

We will write the q-deformed vertex operators only in terms of the weights

this time:

Vω1(x) =: W4(xv−15)W7(xv−15)W−13 (xv−14) :,

Vω2(x) =: W3(xv−16)W−12 (xv−15) :,

Vω3(x) =: W2(xv−17)W−11 (xv−16) :

Vω4(x) =: W1(xv−18) :,

Vω5(x) =: W−16 (x) :,

Vω6(x) =: W6(xv−2)W−15 (xv−1) :,

Vω7(x) =: W5(xv−3)W−14 (xv−2) :,

Vω8(x) =: W−17 (x) :

The explicit formulas in terms of simple root vertex operators can easily be

obtained from this, but they are too long, because of large numbers of simple

roots on the right hand side of (6.21). They can be easily obtained from

above, using expressions in the appendix.

The number of positive roots of E7 is h+(E7) = 63. The W(E7) algebra

is generated by operators of spins sa = 2, 6, 8, 10, 12, 14, 18. The number of

additional parameters, besides the three momenta α0,1,∞ needed to specify

the 3-point conformal block∑7

a=1(sa − 2) = 63 − 7. This is the number

of normalizable and log-normalizable Coulomb moduli of the bulk theory

in the ms to infinity limit. For finite ms, the number of moduli is larger.

The bulk theory has: 133 − 7 Coulomb moduli. Tho specify the vacuum of

the 3d quiver gauge theory, we need as many parameters as positive roots

counted with multiplicities: 25 roots have multiplicity 1; 21 positive roots

have multiplicity 2 as they have intersection number −1 with two of the

weights; 7 roots have multiplicity 2 as they have intersection −2 with one of

the weights; 16 roots have multiplicity 3, as they have intersection number−1

65

with three of the weights; one positive root has intersection number −1 with

4 weights, and hence multiplicity 4. The count of roots with multiplicities

gives 133, and we have to subtract 7 constraints on the net ranks. This is

the same as the number of Coulomb moduli in the bulk (2, 0). It is also the

same as the number of choices in assigning contour of integration in the free

field formulation of the q-deformed 3-point conformal block of the Wq,t(E7)

algebra.

6.5 E8 with 3 Full Punctures

For the weight system WS , we take

ω1 = −w1 + 7e1 + 13e2 + 19e3 + 16e4 + 12e5 + 8e6 + 4e7 + 10e8

ω2 = −w1 + 7e1 + 13e2 + 20e3 + 16e4 + 12e5 + 8e6 + 4e7 + 10e8

ω3 = −w1 + 7e1 + 14e2 + 20e3 + 16e4 + 12e5 + 8e6 + 4e7 + 10e8

ω4 = −w1 + 8e1 + 14e2 + 20e3 + 16e4 + 12e5 + 8e6 + 4e7 + 10e8

ω5 = −w7

ω6 = −w7 + e7

ω7 = −w7 + e6 + e7

ω8 = −w7 + e5 + e6 + e7

ω9 = −w8

(6.25)

The first 4 weights ωi above are in the Weyl group orbit of w1, the next four

in the one of w7. This can be rewritten as:

ω1 = −w3 + w4 + w8

ω2 = −w2 + w3

ω3 = −w1 + w2

ω4 = w1

ω5 = −w7

ω6 = −w6 + w7

66

ω7 = −w5 + w6

ω8 = −w4 + w5

ω9 = −w8.

The weight system provides a basis for the weight lattice of E8, and the 9

vectors sum up to zero.

6.5.1 The Bulk E8 Gauge Theory

The E8 quiver gauge theory describing the low energy dynamics of the bulk

E8 little string theory on a sphere with three full punctures is given in figure

6. To derive the quiver, we sum up separately the simple roots and the

fundamental weights on the right hand side of (6.25). The simple roots sum

to

[S] = 29e1 + 54e2 + 79e3 + 64e4 + 49e5 + 34e6 + 19e7 + 40e8, (6.26)

whose coefficients are the ranks of the corresponding gauge groups. The

fundamental weights sum up to

[S∗] = −4w1 − 4w7 − w8, (6.27)

where the coefficients encode the ranks of the fundamental flavor groups for

each node of the quiver. With the gauge theory description in tow, we can

compute the partition function of the bulk little string using results in 2.2.

6.5.2 The 3d E8 Gauge Theory

The second way of writing the weight system lets us read off the N = 2

chiral matter content of the 3d quiver gauge theory on the D3 branes. The

number of N = 2 (anti)-chiral multiplets on the a-th node are encoded in

the coefficients of wa in the expansion of ωi in terms of fundamental weights.

This leads to the quiver in figure 12. From the quiver, we can compute the

partition function of the theory as in section 3.3.

67

Figure 12: D3 quiver from E8 little string.

6.5.3 E8 Toda CFT

The q-deformed vertex operator is∏9

i=1 Vωi(xi) where:

Vω1(x) =: W4(xv−27)W8(xv−27)W−13 (xv−26) :,

Vω2(x) =: W3(xv−28)W−12 (xv−27) :,

Vω3(x) =: W2(xv−29)W−11 (xv−28) :

Vω4(x) =: W1(xv−30) :,

Vω5(x) =: W 67 (xv6) :,

Vω6(x) =: W7(xv4)W−16 (xv5) :,

Vω7(x) =: W6(xv3)W−15 (xv4) :,

Vω8(x) =: W5(xv2)W−14 (xv3) :

Vω9(x) =: W−18 (xv3) :

The number of positive roots of E8 is h+(E8) = 120. The W(E8) algebra

is generated by operators of spins sa = 2, 8, 12, 14, 18, 20, 24, 30. The number

of parameters, needed to specify the 3-point conformal block, besides α0,1,∞

is∑8

a=1(sa−2) = 120−8, and this is also the number of parameters one has

to specify to pick out a vacuum of the 3d theory. For finite ms, the number

of moduli is larger. The bulk theory has 368 − 8 Coulomb moduli. The

number of vacua of the 3d quiver gauge theory is now the number of positive

68

roots with multiplicities: 32 positive roots have multiplicity 1; 30 positive

roots have multiplicity 2 since they have intersection number −1 with 2 of

the weights; 32 more have multiplicity 2 as they have intersection number

−2 with one of the weights; 44 have multiplicity 3 as they have intersection

number −1 with 3 weights; 8 positive roots come with multiplicity 3 because

they have intersection number −3 with one weight; 14 positive roots have

multiplicity 4, since they have intersection number −1 with 4 weights. The

net count is 368, and we still need to subtract 8 for the net rank constraints.

Acknowledgments

We are grateful to Tudor Dimofte, Davide Gaiotto, Ori Ganor, Andrei Ok-

ounkov, Nicolai Reshetikhin, Christian Schmidt, Shamil Shakirov, Nathan

Seiberg, Cumrun Vafa and Dan Xie for helpful discussions. The research of

M.A. and N. H. is supported in part by the Berkeley Center for Theoretical

Physics, by the National Science Foundation (award number 0855653), by

the Institute for the Physics and Mathematics of the Universe, and by the

US Department of Energy under Contract DE-AC02-05CH11231.

A Wq,t(g) Algebra and Vertex Operators

Here, we will review the construction of Wq,t(g), the q, t-deformed ADE W-

algebra, its screening charges and vertex operators. The section it taken from

[24], with minor changes of notation, and specializing to the simply laced

case. Consider the Heisenberg algebra, generated by modes of n bosons,

with relations

[ea[k], eb[m]] =1

k(q

k2 − q−

k2 )(t

k2− − t−

k2 )Cab(q

k2 , t

k2 )δk,−m

where Cab(q, t) is a q, t deformed Cartan matrix, Cab(q, t) = (v+v−1)δab−Iab,

69

Iab is the incidence matrix of the Dynkin diagram, and v =√q/t. The

generators ea[m] are called the ”root type” generators in [24]. They also

introduce fundamental weight type generators wa[m]

[ea[k], wa[m]] =1

k(q

k2 − q−

k2 )(t

k2 − t−

k2 )δabδk,−m

satisfying

ea[k] =n∑b=1

Cab(qk, tk)wb[k]. (A.1)

The screening charge operators are

Sa(x) =: exp(∑k 6=0

ea[k]

qk2 − q− k2

ekx)

: .

([24] introduce another set of screening charges that correspond to D3 branes

wrapping a complementary subspace of R4, the one rotated by t instead of

q, and which we won’t need here. Furthermore, for brevity we are cavalier

about the zero modes.) In the limit where R goes to zero, this becomes

Sa(x) =: exp(∑

k 6=0ea[k]k ε1

ekx)

:, which are the usual expressions for screening

charges of bosons ϕa(x) = (ea, ϕ(x)) =∑

k∈Zea[k]kekx. It is easy to see that

the two point functions of the screening charges exactly reproduce (4.14),

(4.15).

We associate primary vertex operators to a collection of n+ 1 weights ωi

such that any n of them provide a basis of the weight space, and∑n+1

i=1 ωi =

0, and in addition we require −ωi to lie in the Weyl orbit of one of the

fundamental weights wa. Then, the corresponding vertex operator is given

by

:n+1∏i=1

Vωi(xi) : (A.2)

where Vωi itself is constructed out of fundamental weight and simple root

70

vertex operators, as in section 6.11

Wa(x) =: exp(∑

k 6=0

wa[k]

(qk2 − q− k2 )(t

k2 − t− k2 )

ekx)

: (A.3)

and

Ea(x) =: exp(∑

k 6=0

ea[k]

(qk2 − q− k2 )(t

k2 − t− k2 )

ekx)

: (A.4)

We can relate the two sets of vertex operators, using the relation (A.1).

To take the limit back to the W(g) algebra we write q = eRε1 , t = e−Rε2

and exi = zeRαi , and take R to zero, while rescaling ea[k] and wa[k] by a power

of R for the Heisenberg algebra to continue making sense. The momentum

α carried by Vα(z) is α =∑n+1

i=1 αi ωi. The individual Vωi(xi) do not have a

good conformal limit, but the products in (A.2) do, (4.2):

:n+1∏i=1

Vωi(xi) : → Vα(z).

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